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Theory of dissipative chaotic atomic transport in an optical lattice V.Yu. Argonov and S.V. Prants Laboratory of Nonlinear Dynamical Systems, Pacific Oceanological Institute of the Russian Academy of Sciences, 43 Baltiiskaya st., 690041 Vladivostok, Russia Westudydissipativetransportofspontaneouslyemittingatomsina1Dstanding-wavelaserfield in the regimes where the underlying deterministic Hamiltonian dynamics is regular and chaotic. A Monte Carlo stochastic wavefunction method is applied to simulate semiclassically the atomic dynamics with coupled internal and translational degrees of freedom. It is shown in numerical 2 experiments and confirmed analytically that chaotic atomic transport can take the form either of 1 ballisticmotionorarandomwalkingwithspecificstatisticalproperties. Thecharacterofspatialand 0 momentumdiffusionintheballisticatomictransportisshowntochangeabruptlyintheatom-laser 2 detuningregime where the Hamiltonian dynamics is irregular in the sense of dynamical chaos. We n findaclearcorrelationbetweenthebehaviorofthemomentumdiffusioncoefficientandHamiltonian a chaos probability which is a manifestation of chaoticity of the fundamental atom-light interaction J in the diffusive-like dissipative atomic transport. We propose to measure a linear extent of atomic 1 clouds in a 1D optical lattice and predict that, beginning with those values of the mean cloud’s momentum for which the probability of Hamiltonian chaos is close to 1, the linear extent of the ] corresponding clouds should increase sharply. A sensitive dependence of statistical characteristics h ofdissipativetransport onthevaluesof thedetuningallows tomanipulatetheatomic transportby p changing thelaser frequency. - m PACSnumbers: 37.10.Vz,05.45.Mt,05.45.-a o t a . I. INTRODUCTION kicked ultracold atoms have been shown to demonstrate s c under appropriate conditions the effect of dynamical lo- i calization in momentum distributions which means the s Anatomplacedinalaserstandingwaveisactedupon y quantum suppression of chaotic diffusion [6–14]. Deco- bytworadiationforces,deterministicdipoleandstochas- h herence due to SE or noise tends to suppress this quan- p tic dissipative ones [1–3]. The mechanical action of light tum effect and restore classical-like dynamics [15–18]. [ upon neutralatoms is atthe heartof lasercooling,trap- Another important quantum chaotic phenomenon with ping, and Bose-Einstein condensation. Numerous appli- 1 cold atoms in far detuned optical lattices is a chaos as- cations of the mechanical action of light include isotope v sisted tunneling. In experiments [19–21] ultracoldatoms separation, atomic litography and epitaxy, atomic-beam 9 have been demonstrated to oscillate coherently between 1 deflection and splitting, manipulating translational and tworegularregionsinmixedphasespaceeventhoughthe 3 internalatomic states,measurementofatomic positions, classical transport between these regions is forbidden by 0 and many others. Atoms and ions in an optical lattice, a constant of motion (other than energy). 1. formed by a laser standing wave,are perspective objects 0 for implementation of quantum information processing Thetransportofcoldatomsinopticallatticeshasbeen 2 and quantum computing. Advances in cooling and trap- observedto takethe formofballisticmotion,oscillations 1 ping of atoms, tailoring optical potentials of a desired in wells of the optical potential, Brownian motion [22], v: form and dimension (including one-dimensional optical anomalous diffusion and Lévy flights [23–27]. The Lévy i lattices),controllingthelevelofdissipationandnoiseare flights have been found in the context of subrecoil laser X nowenablingthedirectexperimentswithsingleatomsto cooling [23, 24] in the distributions of escape times for r study fundamental principles of quantum physics, quan- ultracold atoms trapped in the potential wells with mo- a tum chaos, decoherence, and quantum-classical corre- mentum states close to the dark state. In those experi- spondence (for recent reviews on cold atoms in optical mentsthevarianceandthe meantimeforatomstoleave lattices see Ref. [4, 5]). the trap have been shown to be infinite. Experimental study of quantum chaos has been car- A new arena of quantum nonlinear dynamics with ried out with ultracold atoms interacting in δ-kickedop- atomsinopticallatticesisopenedifweworkneartheop- tical lattices [6–13]. To suppress spontaneous emission tical resonance andtake the dynamics of internalatomic (SE) and provide a coherent quantum dynamics atoms states into account. A single atom in a standing-wave in those experiments were detuned far from the optical laser field may be treated as a nonlinear dynamical sys- resonance. Adiabaticeliminationoftheexcitedstateam- tem with coupled internal (electronical) and external plitude leads to an effective Hamiltonian for the center- (mechanical) degrees of freedom [28–30]. In the semi- of-mass (CM) motion [14], whose 3/2 degree-of-freedom classical and Hamiltonian limits (when one treats atoms classical analogue has a mixed phase space with regular aspoint-likeparticlesandneglectsSEandotherlossesof islandsembeddedinachaoticsea. DeBrogilewavesofδ- energy), a number of nonlinear dynamical effects have 2 been analytically and numerically demonstrated with section. Anomalous statistical properties of dissipative this system: chaotic Rabi oscillations [28–30], Hamil- chaotic walking are quantified and discussed in Sec. V. tonian chaotic atomic transport and dynamical fractals Whereas Secs. IV and V are devoted to solving the first [31–33],Lévyflightsandanomalousdiffusion[30,34,35]. taskof this paper, inSec. VI we considerthe problemof These effects are caused by local instability of the CM manifestations of dynamicalinstability and Hamiltonian motion in a laser field. A set of atomic trajectories un- chaos in dissipative atomic transport. We demonstrate der certain conditions becomes exponentially sensitive analytically and numerically that character of diffusion to small variations in initial quantum internal and clas- ofspontaneouslyemittingatomschangesqualitativelyin sical external states or/and in the control parameters, the detuning regime where the underlying Hamiltonian mainly, the atom-laser detuning. Hamiltonian evolution dynamics is chaotic. To observe this effect in a real ex- isasmoothprocessthatiswelldescribedinasemiclassi- periment with cold atoms in a one-dimensional optical cal approximation by the coupled Hamilton-Schrödinger latticeweproposetomeasurethelinearextentofatomic equations. A detailed theory of Hamiltonian chaotic cloudswithdifferentvaluesoftheirmeanmomentumand transport of atoms in a laser standing wave has been de- predictthat the extentshouldincreasesignificantly with veloped in our recent paper [33]. those values of the mean momentum for which the un- The aim of the present paper is to study dissipative derlying Hamiltonian evolution is chaotic. chaotic transport of atoms in a one-dimensional optical lattice in the presence of SE events which interrupt coherent Hamiltonian evolution at random time instants. II. MONTE CARLO WAVEFUNCTION Generally speaking, deterministic (dynamical) chaos is MODELING OF THE ATOMIC DYNAMICS practicallyindistinguishableinsomemanifestationsfrom a random (stochastic) process. The problem becomes much more complicated when noise acts on a dynamical In the frame rotating with the laser frequency ω , the f system which is chaotic in the absence of noise. Such standard Hamiltonian of a two-level atom in a strong systems are of a great practical interest. Comparatively standing-wave 1D laser field has the form weak noise may be treated as a small perturbation to deterministic equations of motion, and one can study Pˆ2 1 in which way the noise modifies deterministic evolution Hˆ = + ~(ω −ω )σˆ − a f z 2m 2 on different time scales. However, SE is a specific shot a (1) Γ quantum noise that cannot be treated as a weak one be- −~Ω(σˆ−+σˆ+)coskfXˆ −i~ σˆ+σˆ−, causetheinternalstatemaychangesignificantlyafterthe 2 emission of a spontaneous photon. Special methods are needed to describe correctly the dynamics of a sponta- whereσˆ±,z arethePaulioperatorsfortheinternalatomic neously emitting single atom in an optical lattice. The degreesoffreedom,Xˆ andPˆ aretheatomicpositionand purpose of this paper is twofold. Our first goalis to give momentumoperators,ω ,ω ,andΩaretheatomictran- a f a description of possible regimes of dissipative atomic sition, laser, and Rabi frequencies, respectively, and Γ is transportinthepresenceofSEandtoquantifytheirsta- the spontaneous decay rate. Internal atomic states are tistical properties. Our secondary intent is a search for described by the wavefunction |Ψ(t)i=a(t)|2i+b(t)|1i, manifestations of the fundamental dynamical instability with a and b being the complex-valued probability am- and Hamiltonian atomic chaos in the diffusive-like CM plitudestofindanatomintheexcited|2iandground|1i motionofspontaneouslyemittingatomsinalaserstand- states. Notethatthenormofthewavefunction,|a|2+|b|2, ing wave which can be observed in real experiments. isnotconservedduetonon-HermiteantermintheHamil- The paper is organized as follows. In Sec. II we for- tonian. mulate a Monte Carlo stochastic wavefunction approach We use the standard Monte Carlo wavefunction tech- to solving semiclassical Hamilton-Schrödinger equations nique[36]tosimulatetheatomicdynamicswiththecou- of motion for a two-level atom in a one-dimensional pled internal and external degrees of freedom in an opti- monochromatic standing light wave. This approach al- cal lattice. The evolution of an atomic state |Ψ(t)i con- lows to get the most probabilistic outcome that can be sists of two parts: (i) jumps to the ground state (a = 0, compareddirectly with corresponding experimental out- |b|2 = 1) each of which is accompanied by the emission put with single atoms. In Sec. III we review briefly of an observable photon at random time moments with our previous results on Hamiltonian chaotic CM motion the mean time 2/Γ (actually, the probability of SE de- whichare necessaryto quantify andinterpretdissipative pends on the atomic population inversion) and (ii) co- dynamics. Sec. IV is devoted to description of possi- herentevolutionwith continuously decayingnormof the ble regimes of dissipative CM motion of spontaneously atomic state vector without the emission of an observ- emitting atoms in a standing wave. Monte Carlo sim- able photon. The decaying norm of the state vector is ulations of the well-known effects of acceleration, decel- equal to the probability of spontaneous emission of the eration, and velocity grouping, and of a novel effect of next photon. It is convenient to introduce the new real- dissipative chaotic walking of atoms arepresentedinthis valued variables (normalized all the time) instead of the 3 amplitudes a and b (renormalized after SE events only) whichwillbeusedinthenextsections. Intheabsenceof any losses (γ =0) the total atomic energy is conserved, 2Re(ab∗) −2Im(ab∗) |a|2−|b|2 u≡ |a|2+|b|2, v ≡ |a|2+|b|2 , z ≡ |a|2+|b|2, H ≡ ωrp2 +U, U ≡−ucosx− ∆z. (5) (2) 2 2 which have the meaning of synphase and quadrature The correspondinglossless equations of motionwith two components of the atomic electric dipole moment and independent integrals of motion, the energy H and the thepopulationinversion,respectively. Westressthatthe length of the Bloch vector, have been shown [28, 29] to lengthoftheBlochvector,u2+v2+z2 =1,isconserved. be chaotic in the sense of an exponential sensitivity to Sincewestudymanifestationofquantumnonlinearef- small variations in initial conditions and/or the control fects in ballistic transport of atoms, when the average parameters. The CM motion is governed by the simple atomic momentum is very large as compared with the equationfor anonlinearphysicalpendulum withthe fre- photon momentum ~k , the translational motion is de- f quency modulation [39] scribed classically by Hamilton equations. The whole atomic dynamics is governedby the following Hamilton- x¨+ω u(τ)sinx=0, (6) r Schrödinger equations [42, 43] where the synchronized component of the atomic dipole ∞ u is a function of all the other atomic variables includ- x˙ =ω p, p˙ =−usinx+ ρ δ(τ −τ ), r j j ing the translational ones. Besides the regular CM mo- Xj=1 tion,namely,oscillationsinawelloftheopticalpotential ∞ γ and a ballistic motion over its hills, we have found and u˙ =∆v+ uz−u δ(τ −τ ), j 2 quantified chaotic CM motion [28, 29, 39]. On the exact Xj=1 (3) atom-laser resonance with ∆ = 0, u is a constant, and ∞ γ the CM performs either regular oscillations, if H < |u|, v˙ =−∆u+2zcosx+ vz−v δ(τ −τ ), j 2 or moves ballistically, if H >|u|. Xj=1 At ∆ 6= 0, the depth of the potential wells changes in ∞ z˙ =−2vcosx− γ(u2+v2)−(z+1) δ(τ −τ ), course of time, and atoms may wander in a rigid optical j 2 lattice (without any modulations of its parameters) in a Xj=1 chaotic way with alternating trappings in the wells and where x ≡ k hXî and p ≡ hPî/~k are normalized flights of different lengths and directions over the hills. f f At small detunings, |∆| ≪ 1, the second term of the atomic CM position and momentum. The dot denotes potential energy U in Eq. (5) is small, and U can be differentiation with respect to the normalized time τ ≡ approximatedby a function of only one internalvariable Ωt. Throughout the paper we fix the values of the nor- u. In this case we have approximate solutions for v and malized decay rate γ ≡ Γ/Ω and the recoil frequency ω ≡~k2/m Ω to be γ =3.3·10−3 andω =10−5. This z r f a r values are similar to those used in experiments with Na τ [6, 7], Cs [9, 40] and Rb [12] cold atoms in a standing- v(τ)=± 1−u2 cos2 cosxdτ′+χ0, wave laser field with the maximal Rabi frequency of the Z p orderof1÷5GHz. So,thenormalizeddetuningbetween  0  (7) τ the field and atomic frequencies, ∆ ≡ (ω −ω )/Ω, is a f a single variable parameter. Also we fix the initial condi- z(τ)=∓ 1−u2 sin2 cosxdτ′+χ0, Z tions as follows: x = v = u = 0,z = −1, and vary p 0 0 0 0  0  theinitialmomentump only. InEqs. (3)τ arerandom 0 j timemomentsofSEeventsandρj arerandomrecoilmo- where χ0 is an integration constant which is a function mentawiththevaluesbetween±1(1Dcase). Intermsof ofinitialvaluesofz andu. Usingthesesolutionsonecan thenormalizedtimeτ therateofoccurrenceofSEevents prove that at |∆| ≪ 1, u performs shallow oscillations is γ(z+1)/2. At τ =τ , the atomic variables change as whentheatommovesbetweenthenodes(recallthatu= j follows: constat∆=0). Theseoscillationsaresynchronizedwith the oscillations of z, and when an atom approaches any τ =τ ⇒u→0, v →0, z →−1, p→p+ρ , −1≤p ≤1. node with cosx=0,where the strengthof the laserfield j j j (4) changes the sign, they slow down (see Eq. (7)). The swingofoscillationsofugraduallyincreases,andexactly at the node u changes abruptly its value (see Fig. 1). III. A BRIEF REVIEW OF HAMILTONIAN Thus, u is practically a constant between the nodes and ATOMIC DYNAMICS it performs a sudden jump at every node. In the Raman-Nath approximation, where x = ω pτ r In this section we review briefly our main results on and p = const, we have managed to derive the deter- Hamiltonian atomic dynamics (see Refs. [28–31, 33, 35]) ministic mapping allowing to compute the value u just m 4 after crossing the mth node probability density functions (PDFs) for the flight and trapping events in the diffusive approximation: ∆ π 2 π ∞ um ≃sinq1−u2m−1 (cid:20)rωrp(cid:18)v0cos(cid:18)ωrp − 4(cid:19)+ Pfl(l)≃ arQcs(iDn3u)H Xj=0(j+1/2)2∗ +(−1)mz0sin(cid:18)ω2rp − π4(cid:19)(cid:19)+(−1)mz0(cid:21)+arcsinum−1(cid:27), ∗exp−(j+arc1s/i2n)22Hπ2Dul, (8) ∞ (10) Q(D ) where v0 and z0 are the values of v and z at the antin- P (l)≃ u (j+1/2)2∗ odes of the standing wave at x = πk, k = 0,1,2,.... tr arccos3H Xj=0 They are the same at all the antinodes because in the −(j+1/2)2π2D l Raman-Nath approximation v and z are periodic func- ∗exp u . tions of x (see solution (7)). Formula (8) describes the arccos2H series of jumps of two alternating magnitudes (for odd Here Q is a constant, D = ∆2π/4ω p is a diffusion u r node andevenm). Strictlyspeaking,(8)isvalidwithfastbal- coefficientintheuspace. Forcomparativelysmallvalues listic atomsandnotona verylongtime scale. Deviation of l (i. e., with short flights and trappings), we get from of the analytic calculations with Eq. (8) from the exact Eq. (10) the power decay numerical results is demonstrated in Fig. 1a where we plot the function u(τ) for a fastatom with p0 =1900. It Pfl ∝Ptr ∝l−1.5, (11) is obvious that the signal is rather regular but the magnitude of the jumps changes slowly because the Bloch whereas for large l the decay is exponential. Numerical componentsv andz arenotstrictlyperiodicfunctionsof simulationoftheHamiltonianequationsofmotionagrees time. wellwiththe analyticalresults(10)indifferentrangesof Figure 1b plots u(τ) in the regime of Hamiltonian thedetunings. AtypicalPDFfortheflightandtrapping chaotic walking. To quantify chaotic jumps of u we pro- events decaysinitially algebraicallyandhas an exponen- posed in Ref. [33] the stochastic map tial tail. The length of the initial power-law segment is inverselyproportionaltothevalueofthedetuning∆and π can be rather large. um ≃sin(cid:18)|∆|rωrpnode sinφm+arcsinum−1(cid:19), In which way SE changes the statistical properties of (9) the Hamiltonian motion? Can we find fingerprints of which was derivedfrom the deterministic map (8) by in- Hamiltonianinstabilityandchaosinthe motionofspon- troducing random phases φ (0 ≤ φ ≤ 2π) instead of taneously emitting atoms or SE totally suppresses any m arguments of trigonometric functions which may differ manifestations of coherent (but chaotic!) Hamiltonian significantlyfromnodetonodeduetostrongvariationsin dynamics? Thesequestionswillbe addressedinthe next the atomicmomentum beyondthe Raman-Nathapprox- sections. imation. Note that the value of the momentum at the instantwhenthe atomcrossesa node, p = 2H/ω , node r IV. DISSIPATIVE ATOMIC TRANSPORT IN A is approximately the same for all nodes. p LASER STANDING WAVE The map (9) describes a random Markov process in the u space with u varying in the range −1≤u ≤1. m m This quantity may be smaller or larger than the atomic Theemissionofaphotonintothecontinuumofmodes energyH (whichisaconstantinthe Hamiltonianlimit). ofthe electromagneticfieldis accompaniedbyanatomic Since the values of u define the atomic potential energy, recoil. The dissipative (friction) force F ≡ hp˙i (which itsrandomwalkinggovernsarandomwalkingofatomsin does not exist in the Hamiltonian system) depends on the lattice. The possibleregimesofthe HamiltonianCM the atomic momentum p and the sign of the detuning in motioncanbesummarizedasfollows[33]: At|u|>H,an a complicated way [2, 41]. The effects of acceleration, atomoscillatesinoneofthepotentialwells,at|u|<H,it deceleration, and velocity grouping (at ∆ < 0) are well- moves ballistically. It can walk chaotically if 0<H <1. knowninthe literature[2,3]. Anoveleffectwereportin In the process of Hamiltonian chaotic walking the atom this sectionis dissipative chaotic walking. It appears un- wanders in an optical lattice with alternating trappings der appropriate conditions that are different from those in wells of the optical potential and flights over its hills specified for Hamiltonian chaotic walking in the preced- changing the direction of motion many times. “A flight” ing section. is an event when the atom passes, at least, three nodes. To illustrate the possible regimesofdissipative atomic CMoscillationsinawelloftheopticalpotentialiscalled transport in a standing wave we integrate by the Monte “a trapping”. The number of node crossings l during a Carlo method dissipative equations of motion (3) with single flight or a single trapping event is a discrete mea- 2000atomswhosepositionsandmomentaaredistributed sure of the length and durations of those events. We in a quasi-Gaussian manner (Fig. 2a). In Fig. 2b we have derived in Ref. [33] the following formulas for the demonstratethevelocitygroupingeffectat∆=−0.2and 5 00 --00..0022 uu ((aa)) --00..0044 ΛΛ==00,, γγ==00 --00..0066 00 11000000 22000000 33000000 44000000 55000000 66000000 77000000 00..0022 ((bb)) uu 00 ΛΛ==11,, γγ==00 00 11000000 22000000 33000000 44000000 55000000 66000000 77000000 00..0022 ((cc)) 00 uu --00..0022 ΛΛ==00,, γγ==00..00003333 --00..0044 00 11000000 22000000 33000000 44000000 55000000 66000000 77000000 000...000222 222...444777 (((ddd))) 000 ---000...000222 222...444666 ---000...000444 uuu HH ---000...000666 ---000...000888 222...444555 ---000...111 ΛΛΛ===111,,, γγγ===000...000000333333 ---000...111222 000 111000000000 222000000000 333000000000 444000000000 555000000000 666000000000 777000000000 ττ FIG. 1. Time evolution (τ is in units of Ω−1) of the synphase component of the electric dipole moment u and the atomic energyH (inunitsof~Ω). (a)RegularHamiltoniandynamics(p0=1900,γ =0),(b)chaoticHamiltoniandynamics(p0 =700, γ =0), (c) regular dissipative dynamics (p0 =1900, γ =0.0033), (d) chaotic dissipative dynamics (p0 =700, γ =0.0033). In all thepanels, ∆=−0.0005. The initial part of (a) agrees with approximate solution (8) with v0 =0, z0 =−1. τ =105. Alargenumberofatomsisgroupedaroundtwo atoms are accelerated and slow ones are decelerated. As values of the capture momentum p ≃ ±1300 because aresult,weobserveapronouncedpeakaroundx≃p≃0 cap ofaccelerationofslowatomsanddecelerationofthe fast shown in Fig. 2c at ∆=0.1 and τ =105. onesintheinitialensemble. Theslowertheatomsarethe longer is the process of the velocity grouping. Note that Dependence of the friction force F on the current atoms with |p| . 100, trapped initially in a well of the atomic momentum p is shown in Fig. 3 at ∆ =−0.2. It optical potential, could not quit the well up to τ = 105. hasbeencomputedwithourmainequations(3)whenav- Contrary to that, at positive values of the detuning fast eragingoverseventhousands atomswith different initial momentum. The function F(p) resembles the behavior 6 FIG. 2. Atomic momentum and position distribution illustrating the effects of atomic acceleration, deceleration, and the velocity grouping: (a) τ =0, (b) τ =105, ∆=−0.2, (c) τ =105, ∆=0.1, (d) τ =105, ∆=−0.05. Momentum p is given in unitsof ~k , theposition in units of k−1. f f ofthefrictionforcecomputedwithanothermethods(see sult, atoms may change their direction of motion in an [2]andFig. 1ain[41]). Thefrictionforcedecreasesupto irregular way. Such a dissipative chaotic atomic walking zero value and then begin to oscillate with increasing p. is illustrated in Fig. 2d at ∆ = −0.05 and τ = 105 with It has a number of zeroes (the detailed view is shown in theatomsdistributedwidelyinthephaseplane. Typical Fig. 3b) like the corresponding functions in Refs. [2, 41]. atomictrajectoriesareshowninFig.4inthemomentum Zero values of F correspondto quasistationary values of space. Figures 4a and b illustrate how the friction force the momentum which depend on ∆. Some of them are near the resonance (∆ = −0.001 and ∆ = −0.01) decel- attractorsandatomswithclosevaluesofthemomentum erates atoms with large values of the initial momentum tend to p , another ones are repellors. The attractors down to so small values of the capture momentum when cap and repellors are not deterministic because of a random the dissipative chaotic walking becomes possible. With natureofSE.Thus,anatomswalksrandomlyinthemo- increasing the absolute value of the negative ∆, the cap- mentum space between different values of the capture ture momentum increases and the atom changes rarely momentum p . When it reaches a certain value of the the direction of motion (Fig. 4c with ∆=−0.1). Panels cap capture momentum the atom does not stop in the mo- dande inFig.4illustratethe velocitygroupingeffectat mentum space and goes on to fluctuate because of both ∆=−0.15withdifferentvaluesoftheinitialmomentum. the Hamiltonian instability and SE effect. Inthe precedingsectionwedescribedthe Hamiltonian V. STATISTICAL PROPERTIES OF chaotic walking that may occur in the absence of any DISSIPATIVE CHAOTIC WALKING losses. Dissipation causes additional strong fluctuations of the momentum. If ∆ > 0 or if it is negative but comparativelylarge,nothing principally new happens to Statistics of Hamiltonian chaotic walking is quanti- atoms as compared with the Hamiltonian limit. How- fied by the flight and trapping PDFs (10) with alge- ever, at negative small values of ∆, a characteristic cap- braically decaying head segments and exponential tails ture momentum becomes smaller than a typicalrangeof whose lengths strongly depend on ∆. We will show in momentum fluctuations due to atomic recoils. As a re- this section that PDFs for dissipative chaotic walking 7 250 00..11 p 0 (a) ∆ = −0.001 -250 00..0088 0 0.25 0.5 0.75 1 750 ∆ = −0.01 .. 00..0066 500 <<pp>> 250 00..0044 0 -250 (b) 00..0022 -500 0 0.25 0.5 0.75 1 00 1750 ∆ = −0.1 1500 00 550000 11000000 11550000 22000000 1250 1000 00..000022 750 500 250 0 00 -250 -500 -750 (c) -1000 --00..000022 0 0.25 0.5 0.75 1 1750 (d) ∆ = −0.15 11110000 11220000 11330000 11440000 1500 1250 pp 1000 750 FIG. 3. Dependence of the friction force F ≡ hp˙i on the 500 current atomic momentum p at thedetuning∆=−0.2. 0 0.25 0.5 0.75 1 1500 (e) 1250 1000 are even more sensitive to variations in ∆. Figures 4a 750 and b clearly demonstrate that at very small detuning 500 ∆=−0.001longflightsdominate,whereasthereappears 250 anumberofshortflightswithlargervalueof∆=−0.01. 0 ∆ = −0.15 InFig.5weplotthePDFsP forthedurationofatomic flights T with different valufels of the detuning ∆. At -2500 0.25 0.5 0.75 1 x106 small detunings (Fig. 5a), the length of the power-law τ segmentsdependson∆inasimilarwayasinthe Hamil- tonian case (compare this figure with Figs. 5a, 6a, and FIG. 4. Typical atomic trajectories in the momentum space: 7a in Ref. [33]). However, the slope slightly differs from (a)-(c) dissipative chaotic walking with different statistics of the Hamiltonian slope which is equal to −1.5. The dif- atomic flights, (d)-(e) the effect of velocity grouping. Note ference in the statistics of dissipative and Hamiltonian that atoms with very different initial momentum acquire a close valueof thecapturemomentum. walkings is more evident with larger values of the detuning (Fig. 5b). The length of the power-law segments increasesdrasticallywithincreasing∆. This effectisab- sent in Hamiltonian dynamics. The corresponding slope medium values −0.12≤∆≤−0.06. Both the quantities α decreaseswith changingthe detuning from∆=−0.09 correlate well with each other. It means that, changing to ∆ = −0.12 because of the corresponding increase in the value of ∆, one can manipulate statistical properties the length of atomic flights (see Figs. 4a, b, and c). In ofdissipativeatomictransportinanopticallattice. This Fig. 6 we plot the dependencies of the mean duration controlisnonlinearinthesensethatslightlychanging∆, of atomic flights hTi and the slope of the PDF power- say,from−0.08to −0.12,weincreasethe meanduration law fragments α on the detuning ∆ in the range of its of flights in a few orders of magnitude. This effect may 8 ----4444 77 33 ----6666 >> 66 22 lolo PPPPflflflfl <T<T 11gg 10101010 00 00(( ogogogog ----8888 gg11 55 11 -1-1 llll // oo αα ll )) ----11110000 ((((aaaa)))) 44 00 ----11112222 4444 5555 6666 7777 8888 33 --11 --00..1122 --00..11 --00..0088 --00..0066 ∆∆ ------444444 FIG.6. Dependenciesofthelogarithmsofthemeanduration ------666666 ofatomicflightshTi(solidline)andoftheslopeαofthePDF PPPPPPflflflflflfl power-lawfragments(squares)onthedetuning∆(inunitsof 000000 Ω). 111111 gggggg ------888888 oooooo llllll terval τj < τ < τj−1. The real energy H (see Fig. 1d) ------111111000000 ((((((bbbbbb)))))) decreases a little in between in a linear way. The rate of thisdecreaseisdefinedbythecoefficientsofspontaneous emission γ, the detuning ∆, and the average probability tofindtheatomintheexcitedstate. Boththequantities, ------111111222222 444444 555555 666666 777777 888888 H andH˜,changesabruptlyjustafterSE(becauseofthe correspondingchanges in the atomic variables (4)). Just lllllloooooogggggg TTTTTT 111111000000 after emitting a jth spontaneous photon at τ =τ , they j have the same values. So, we will model the evolutionof FIG.5. ThePDFsPflforthedurationofatomicflightsT with the energy as a map Hj ≡H(τj+) taken at the moments (a) small detunings (crosses ∆ = −0.01, stars ∆ = −0.001, τ+ just after SE j circles∆=−0.0001,squares∆=−0.00001)and(b)medium dαet=un−in0g.2s7(;stsaqrusar∆es=∆−=0.−090,.1α2,=α−=0.−770;.0c5i)r.cleSstr∆aig=ht−li0n.e1s, Hj =H˜j −H˜j−1+Hj−1 =Hj−1+ωrp(τj−)ρj+ slohgo-wlogslsocpaeles.α of the power-law fragments of the PDFs in + ω2rρ2j + ∆2 +u(τj−)cosx(τj)+ (13) ∆ ∆γ + 2z(τj−)+ 4 h1−z2i(τj −τj−1), be qualitatively explained as follows. When increasing − − the absolute value of the negative detuning, the capture where the values of the atomic variables p(τj ),u(τj ), − − momentum increases but fluctuations of the currentmo- and z(τ ) are taken at the moments τ just before SE. j j mentum p decrease providing long atomic flights [43]. They are, in turn, determined by the coherent evolution To explain the statistical properties of the dissipa- between SE. tive chaotic walking let us consider the behavior of the Thestochasticmapfortheatomicenergy(13)provides quasienergy an important information about the CM motion. It has been shown in Ref. [33] that atoms move ballistically, H˜ ≡ ωrp2−ucosx− ∆z− ∆γh1−z2i(τ −τ )= if the atomic energy satisfies to the condition H & |u|, j 2 2 4 j whereasatH .|u|theymaychangethedirectionofmo- ∆γ tion. The dissipative chaotic walking takes place when =H − 4 h1−z2i(τ −τj), τj <τ <τj−1, theatomicenergyH alternativelytakesthevalueslarger (12) and smaller than a critical value H =|u|. In the Hamil- which is equal to the total atomic energy (5) in the ab- tonian limit, where the energy is conserved,the problem sence of relaxation. Near the resonance, |∆| ≪ 1, H˜ of the CM chaotic walking has been reduced to the task j is almost conserved between SE events, i. e., in the in- ofrandomwalkingoftheBlochcomponentu(seeSec. III 9 and Ref. [33]). The energy is not conserved in the pres- Using weak Raman-Nath approximation, (32) and (34), ence of relaxation, but the values of u are always small the first term can be replaced by γω H/6. Using the r (see Appendix and Fig. 1c and d). Thus, atoms oscillate estimation (38) for hu2(τ−)i in the irregular CM motion j in the wells of the optical potential if H . 0 and move regime (see Appendix), we get the following expression ballistically if H &0. for the energy diffusion coefficient: On a time scale exceeding the mean time between SE events 2/γ, the evolution of energy can be treated as a Dch ≃ γωrH + ∆2. (19) diffusion process with a drift in the energetic space. The H 6 8 probability to have the energy H at time τ is governed Thisexpressionisvalidformoderatelysmallmomentums by the Fokker-Planck equation (p.1000)when the strong Raman-Nath approximation ∂P ∂2P cannot be applied. In the process of dissipative chaotic P˙(H,τ)=−2c +D , (14) H∂H H∂H2 walking, the probability to get higher values of the mo- mentums is almost zero. where DH is an energy diffusion coefficient and cH is an Now we will try to derive analytically a distribution energy drift coefficient which can be estimated with the of the durations T of atomic flights in the process of help of Eq. (13) as follows: dissipative chaotic walking. In fact, it is a problem of the first passage time for the atomic energy H to return c ≡ hHj −Hj−1i =hH˙i≃ ωrγ + ∆γ. (15) to its zero value. Recall that at small detunings we have H hτj −τj−i 12 2 H ≃0 in the very beginning of every flight. In course of time H can reach rather large values, and it returns to In deriving this formula we adopt the average value of zero at the end of the flight. If the random jumps of the the squared recoil momentum hρ2i = 1/3 (the projec- j energy would be symmetric (c = 0), the probability tions of the recoilmomenta onthe standing-waveaxis x, H to find a flight with duration T would be proportional ρj, are assumed to be distributed in the range ±1 with to T−1.5, where the exponent −1.5 does not depend on the same probability), the average value of the popula- − thediffusioncoefficient. Thisconjecturefollowsfromthe tion inversion just before a SE event to be z(τ )=1/2, j knowntheoreminprobabilitytheory. Moregeneralresult the average value of z over the whole time scale and (see chapter XIV in Ref. [44]) proves that in the case its mean squared deviation from 1 to be hzi = 0 and of an asymmetric randomwalking in the energetic space h1−z2i=1/2,respectively (see the solution(33) inAp- (c 6=0)thePDFfortheflightdurationsinconfiguration pendix). Moreover, neglecting the correlation, we put H space is hucosxi ≃ huihcosxi = 0, which is valid if |ω p| & γ/2, r Si.inec.e, twhheefinrsptt&er1m00inw(1it5h)iosusrmcahloliacnedomf tahyebpeanreagmleectteerds,. Pfl ∝e−c2HT/DHT−1.5, (20) the drift velocityofanatominthe energetic spaceis ap- ifthedriftanddiffusioncoefficientsintheFokker-Planck proximately proportionalto the detuning ∆, and, there- equationfor the randomwalking are assumed to be con- fore in averageatoms accelerateand decelerateat ∆>0 stants. This formula gives a distribution of the flight and ∆ < 0, respectively, as it should be for |∆| ≪ 1. In durations with a power-law fragment followed by an ex- theweakRaman-Nathapproximation,(32)and(34),the ponentialtailandagreesqualitativelywiththeexactnu- driftcoefficientintheenergeticspaceissimplyrelatedto merical computations of P shown in Fig. 5a for a few fl the friction force F acting upon atoms values ofthe detuning ∆. The main disadvantageofthis hH˙i formula is that (20) does not depend on ∆ as the ex- F ≡hp˙i≃ . (16) act PDFs in Fig. 5a. At very small ∆ = −10−5, the ω p r PDF, shown by squares in Fig. 5a, decays mostly alge- braically, whereas at larger values of the detuning the Thefrictionforceplaystheroleofadriftcoefficientinthe power-law fragments are much shorter. A discrepancy momentum space. Strictly speaking, the weak Raman- between the analytical and numerical PDFs arises be- Nath approximation is not valid near the turning points cause we assumed in deriving (20) that D and c do when the atomic velocity is comparatively small. How- H H not depend on the energy H. In fact, it is not the case ever, most of the flight time it is valid. forsmallvaluesofthemomentump,andamoreaccurate The diffusion coefficientinthe energetic spaceis given formula for P (T) is required. by the formula fl The PDFs for Hamiltonian (10) and dissipative (20) DH ≡ h(Hj −Hj−1)2i−hHj −Hj−1i2, (17) tproawnesrp-olarwt afrreagsmimeinlatrs ifnolltohweesdenbsye tehxaptonbeontthiaPlftlaiclos,ntbauint 2hτj −τj−1i the origin of each statistics is different. In the Hamilto- which can be rewritten with the help of (13) as follows: nianlimit the statistics is governedbythe behaviorofu, not the energy, as in the dissipative system. A turnover γω2p2(τ−) hu2(τ−)iγ from a power law to an exponential decay in the Hamil- DH ≃ r j + j . (18) tonian case is explained by a boundedness of |u| . 1, 12 8 10 whereas in the dissipative system it is explained by a ofallpossible values ofthe atomicenergyH (5) is parti- negative drift of the energy H. Each of the factors pre- tioned in a large number of bins. For a large number of ventsthecorrespondingrandomlywalkingquantitytogo initial conditions (in fact, we change only the initial mo- far away from its critical value (at which the atoms can mentum p keeping the other conditions to be fixed), af- 0 changethedirectionsofmotion)decreasingtheprobabil- teranyjthSEeventwecomputethedifferenceHj−Hj−1 ity of long flights in at exponential way. and the squared difference (Hj −Hj−1)2. They are random values, but their statistics depend on the preceding energy value Hj−1. So we calculate the histograms of VI. MANIFESTATION OF HAMILTONIAN the average values of hHj −Hj−1i and h(Hj −Hj−1)2i CHAOS IN DISSIPATIVE ATOMIC TRANSPORT as functions of energy Hj−1. After that we can compute the energy diffusion coefficient D (17) which, being di- H IntheabsenceofSEtheatomicdynamicscanberegu- videdbyω2p2,yieldsthemomentumdiffusioncoefficient r lar or chaotic depending on the initial conditions and/or D whichisbettertopresentasafunctionofthecurrent p the detuning. In experiments one measures statistical momentum p≃ 2Hj−1/ωr. characteristicsofspontaneouslyemitting atoms. Isthere ThemainresupltinthissectionisillustratedwithFig.7. a correlation between those characteristics and the un- Intheupperleftandrightpanelsthedependenciesofthe derlying Hamiltonian dynamics? Can we find any man- momentum diffusion coefficient D on the current mo- p ifestations of Hamiltonian instability, chaos, and order, mentum p are plotted in a log-log scale for ∆ = −0.01 in the diffusive-like dissipative atomic transport? These and ∆ = −0.0005, respectively. In both the cases, we questions will be addressed in the present section. put γ = 0.0033. These plots should be compared with The common quantitative criterion of deterministic the corresponding lower panels where the probability of chaos,themaximalLyapunovexponentλ,isameasureof the Hamiltonian chaos Λ is plotted against p with γ =0 a divergence of two trajectories in the phase space with (i. e. in the Hamiltonian limit of Eqs. (3)). It is evident close initial conditions [45]. To quantify probability of that the character of the momentum diffusion changes chaos in the mixed Hamiltonian dynamics, when λ = 0 abruptlyatthosevaluesofthecurrentmomentumwhere with some values of p0 and λ>0 with another values of atransitionfromchaosto orderoccursinthe underlying p0, we introduce a probabilistic measure of Hamiltonian Hamiltonian dynamics. Such a turnover takes place in a chaos range of small negative detunings and is a manifestation of the peculiarities of the underlying Hamiltonian evo- Λ≡h2Θ(λ)−1i, (21) lution in the diffusive-like dissipative transport of atoms in a standing-wave laser field. We may conclude that in where Θ(λ) is a Heaviside function (Θ = 0 if λ < 0, spiteofrandomatomicrecoilsduetoSEthechaotic(reg- Θ = 1/2 if λ = 0, and Θ = 1 if λ > 0). The probability ular) dynamics between the acts of SE clearly manifests of Hamiltonian chaos Λ is computed by averagingovera itself in the behavior of the measurable characteristic of large number of atomic trajectories with different values theatomictransport,themomentumdiffusioncoefficient of p . If all the trajectories in the set turn out to be 0 D . The behavior of D in the range of p, where the p p stable, one gets Λ = 0, and if all they are exponentially underlying Hamiltonian evolution is chaotic, is well de- unstable, then Λ = 1. One gets 0 < Λ < 1, if some scribed by the formula (23) with Dch ∼ p2+const (see trajectories in the set are stable but the other ones are p boththe upper panels inFig.7where this dependence is not. The magnitude of Λ is proportional to the fraction shown by dashed lines). of trajectories with positive λs. However,theformula(23)doesnotworkintheregimes To examine manifestations of the underlying Hamilto- when the underlying Hamiltonian dynamics is mixed or niandynamicsindissipativetransportinisconvenientto regular because in deriving it we supposed fully chaotic consider atomic diffusion not in the energetic but in the behaviorofu. Wehavemanagedtoestimateanalytically momentum space. The momentum diffusion coefficient, D intheHamiltonianregularregimeatextremelysmall p which is a measure of momentum fluctuations, can be valuesofthedetuning|∆|≪1andforatomswhosemo- written with the help of (17) and (32) as follows: mentum is so large that we can neglect its fluctuations D γ hu2(τ−)iγ between SE events (the exact Raman-Nath approxima- Dp ≃ ωr2Hp2 ≃ 12 + 8ωr2jp2 . (22) tbieohnawviiotrhoxfu=wωhripcτh)i.sFdiegsucribee1dabiyllutshteradteetsertmheinlaisdtdicerm-laikpe- ping(8)onacomparativelyshorttimescale. TogetDRN Usingtheformula(19),wegetDpintheregimeofchaotic from Eq. (22) we use the expression (37) for u2(τ−)pde- oscillations of the Bloch component u j rived in Appendix Dpch ≃ 1γ2 + 8ω∆r22p2. (23) DpRN ≃ 1γ2 + 8ω∆rp2γπ. (24) ThemomentumdiffusioncoefficientD iscomputedwith Thus, we derived the formulas for the momentum dif- p the main equations (3) in the following way. The range fusioncoefficientD intheregimesofHamiltonianchaos p