0805.1948v1[1]

Avalanches of Bose-Einstein Condensates in Leaking Optical Lattices G. S. Ng 1 , H. Hennig 2,3 , R. Fleischmann 2 , T. Kottos 1,2 , and T. Geisel 2,3 1 Department of Physics, Wesleyan University, Middletown, Connecticut 06459, USA 2 MPI for Dynamics and Self-Organization, Bunsenstraß e 10, D-37073 Göttingen, Germany 3 Institute for Nonlinear Dynamics, University of Goettingen, 37073 Goettingen, Germany One of the most fascinating experimental achievements of the last decade was the real- ization of Bose-Einstein Condensation (BEC) of ultra-cold atoms in optical lattices (OL's) [1, 2, 3, 4]. The extraordinary level of control over these structures allows us to investigate complex solid state phenomena [4, 5, 6, 7, 8, 9] and the emerging field of �atomtronics� promises a new generation of nanoscale devices. It is therefore of fundamental and technological importance to understand their dynamical properties. Here we study the outgoing atomic flux of BECs loaded in a one dimensional OL with leaking edges, using a mean field description provided by the Discrete Non-Linear Schrodinger Equation (DNLSE). We demonstrate that the atom population inside the OL decays in avalanches of size J . For intermedi- ate values of the interatomic interaction strength their distribution P (J) follows a power law i.e. P(J) ∼ 1/Jα characterizing systems at phase tran- sition. This scale free behaviour of P(J) reflects the complexity and the hierarchical structure of the underlying classical mixed phase space. Our results are relevant in a variety of contexts (when- ever DNLSE is adequate), most prominently the light emmitance from coupled non-linear optics waveguides [12]. An optical lattice with a controlled leakage of the atomic BEC can be realized experimentally by the action of two separate continous microwave or Raman lasers to locally spin-flip BEC atoms (at the edges of the OL) to a nontrapped state [10, 11]. The spin-flipped atoms do not experience the magnetic trapping potential and are released through gravity in two atomic beams at the ends of the OL. The mathematical model (see Method section) that describes the dynamics of the BEC in a leaking OL of size M is i ∂ψn ∂τ = χ |ψn|2 ψn − 12 [ψn−1(1− δn,1) + ψn+1(1− δn,M )] − iγψn[δn,1 + δn,M ]; n = 1, · · · ,M (1) where γ describes the atomic losses and Nn = |ψn|2 is the atomic population at site n. Below we study the decay (due to leakage) of the total atom population N(τ) = ∑ nNn(τ) inside the OL as a function of the ini- tial effective interaction strength i.e. Λ = χN(τ = 0)/M . An exciting result appearing in the frame of nonlin- ear lattices is the existence of stationary, spatially local- ized solutions, termed Discrete Breathers (DB), which emerge due to the nonlinearity and discreteness of the system. DBs were observed in various experimental se- tups [15, 16, 17, 18, 19, 20] while their existence and stability were studied thoroughly during the last decade [21, 22, 23, 24, 25]. Their importance was already recog- nized in [26] where it was shown that they act as virtual bottlenecks which slow down the relaxation processes in generic nonlinear lattices [25, 26, 27]. Further works [27] established the fact that absorbing boundaries take generic initial conditions towards self-trapped DB's. Re- cently the same scenario was proposed for a BEC in a leaking OL where it was observed [10] that N(τ) decays in sudden bursts J (see inset of Fig. 1). Here, for the first time we present a full theoretical study of the decay process of N(τ) and analyze the distribution P(J). We find three types of dynamical behaviour [14]: For very weak interactions Λ, the population decay is a smooth process which does not involve any avalanches. As inter- actions are increased, avalanches are created. For strong interaction strength, their distribution P (J) is exponen- tial. In contrast, for intermediate values of the interac- tion strength, P (J) follows a power law indicating the existence of a phase transition. Below, we will focus our analysis in this critical regime. Since we are interested in the effects of DBs on the re- laxation process, we introduce a localization parameter PR which provides a rough estimate of the relative num- ber of sites that are occupied by the remaining atoms in a leaking OL. It is defined as PR(τ) = [N(τ)] 2 M ∑ n |ψn(τ)|4 (2) which in case of γ = 0 is the standard participation ratio. Accordingly, the more evenly the atoms spread over the lattice, the closer PR is to a constant of or- der 1. The PR approaches two limiting values [14]: (a) PR = 1/2 corresponding to a random superposition of uncoupled sine waves for U = Λ = 0 (linear regime) and (b) PR = 5/9 corresponding to a configuration of atoms trapped in M uncoupled (T → 0 ) wells for Λ → ∞, which might be viewed as the formation of O(M) DB's (multi-breather regime). In the closed system, the tran- sition between these limiting cases is smooth (see lower left inset of Fig. 1). In the open system (γ > 0) PR is a time dependent quantity. We numerically study the value PRS=PR(τ∗) for large times τ∗ when the system has evolved from its initial thermalized state into a quasi-steady state [31]. For nonzero γ, instead of a smooth transition between the two extremes we observe a sharp drop of the PRS at a critical interaction strength of Λb ≈ 0.15 resembling a ar X iv :0 80 5. 19 48 v1 [ co nd -m at. sta t-m ec h] 1 4 M ay 20 08 2 phase transition (our numerics indicate that this transi- tion becomes a step function in the limitM →∞). PRS drops down to its lowest possible value (∝ 1/M) corre- sponding to a single occupied site, i.e. the final state consists of a one self-trapped DB. If Λ increases further (Figs. 2b-c) the final state consists of a successively larger number of breathers, while for some critical Λ it evolves into a more complicated state with a power-law distribu- tion of norms Nn (see Fig. 1 left upper inset) [32]. In the following it is important to realize that if a breather solution exists for some value of Λ, it exists for all Λ′s > 0 (and for large enough M). This conclusion can be easily drawn by noting that a self-trapped DB is not directly coupled to the leaking edges, thus we can assume γ = 0 and then appropriately scale Eq. 5. There- fore breather solutions in particular do exist for Λ < Λb as well. So, what is the nature of the sharp transition at Λb? To answer this question, we have to examine the thermalized random initial states more closely. For small Λ ≈ 0 these states have exponentially distributed norms PNn(x) = Mexp(−Mx) [14]. In addition, our numerical calculations (see the phase space analysis below) indi- cate that the minimal total nonlinearity associated with the central site of a self-trapped DB solution satisfy the relation MΛ |ψ|2 ≈ 1 . Thus, a necessary condition in order to excite such a DB is that at least one site of the thermalized initial state has a norm x > x0 = 1/(MΛ). Assuming independent random norms for the individ- ual sites we get that the probability for such an event is WM (x) = 1 − (1 − exp(−Mx))M . By increasing M , we find that WM (x) posseses a steep transition from 1 to 0 at approximately x1/2 with WM (x1/2) = 1/2. We can thus give a rough estimation of the lower bound for Λb by demanding x1/2 ≥ x0, which leads to Λb ≥ − 1 ln ( 1− [0.5]1/M ) ≈ 1 lnM − ln ln 2 (3) in fair agreement with the value Λb ≈ 0.15 for the system sizes we studied [33]. The most interesting situation emerges for large enough system sizes and (for the system sizes we could numerically study) interatomic interactions in the range of Λ ≈ 0.5 − 1 [34]. In this case we have found that the atoms leak out of the lattice in avalanches follow- ing a scale-free distribution, i.e. P(J) ∼ J−α, while the norms Nn at individual lattice sites are distributed as PNn(x) ∼ x−βwith α ≈ β (see Fig. 1 left upper inset). This power law behavior for a whole range of parameter values Λ could be interpreted as a signature of a self- organized critical state [28]. Let us study in more detail the dynamics that lead to the creation of an avalanche in this parameter regime. One such event is depicted in Fig. 2d. A moving DB, coming from the bulk of the lattice, collides with the outer most self-trapped DB. As a result the self-trapped breather is shifted inwards by one site while at the same time a particle density proportional to the density of the moving breather tunnels through the self-trapped DB. Eventually this particle density will reach the leaking edge of the OL and decay in a form of an avalache. We have also found numerically [14] that during the migration process of the self-trapped DB, the number of particles and the energy of the three lattice sites in- volved is conserved, thus allowing us to turn the problem to the analysis of a reduced M = 3 system with inter- action strengths Λ in the critical range. As shown in Fig. 3a (inset), the non-linear trimer exhibits a hierar- chical mixed phase space structure with islands of regular motion (tori) embedded in a sea of chaotic trajectories. Trajectories inside the islands correspond to self-trapped DBs, provided that their frequency is outside the linear spectrum. In contrast, chaotic trajectories have contin- uous Fourier spectra, parts of which overlap with the linear spectrum of the infinite lattice [21]. As long as the self-trapped DB is stable, it acts as a barrier which pre- vents atoms from reaching the leaking boundary. Thus, a necessary condition for an avalanche event is the destabi- lization of the DB. This is caused by a lattice excitation with particle density Npert1 greater than a critical value which can push the regular orbit out of the island. At the same time a portion Nmax3 ∝ Npert1 transmits through (see Fig. 3a). As can be seen from Fig. 2, the migra- tion process usually moves the DB inwards by one site which correspond to another island in the phase space of the extended lattice. Details of the migration process are still under investigation [14, 30]. We conjecture that the size of avalanches J is propor- tional to the size S of islands in the mixed phase space found in the Poincaré section of the non-linear trimer. A heuristic model [14] that mimics the hierarchical (`island- over island') structure of a typical mixed phase space leads us to a power law distribution P (S) ∼ S−α where 1 < α < 3. The lower bound comes from the requirement of having an infinite number of islands (self-similar prop- erty) while the upper-bound results from the requirement that the total volume of the phase space be finite. To ver- ify the above prediction for P (S) is a computationally demanding task. Therefore we used a numerically more convenient model: the kicked rotor which is a paradig- matic model of mixed phase space dynamics [29]. Leav- ing aside the technical details [14], we present in Fig. 3b the outcome of our numerical calculations. The results confirm the power law scaling of the island sizes over more than three orders of magnitude thus confirming the validity of our conjecture. In conclusion, we have shown that there is a critical regime of interatomic interactions, where particles are ejected out of a leaky OL in scale free avalanches. The observed power law distribution, is dictated by the hier- archical structure of the mixed phase space. Our results are quite universal and should be observable in many other experimental realizations of the DNLSE including molecular crystals, globular proteins and non-linear op- tics. 3 METHODS The simplest model that captures the dynamics of a dilute gas of bosonic atoms in a deep OL with chemical potential small compared to the vibrational level spac- ing, is the Bose-Hubbard Hamiltonian. In the case of weak interatomic interactions (superfluid limit) and/or a large number of atoms per well (so that the total num- ber of atoms N ∼ O(104 − 105) is much bigger than the number of wells M), a further simplification is available since the BECs dynamics admits a semiclassical mean field description [13]. The resulting Hamiltonian is H = M∑ n=1 [U |ψn|4 +µn|ψn|2]− T2 M−1∑ n=1 (ψ∗nψn+1 + c.c.) (4) where n is the well index, |ψn(t)|2 ≡ Nn(t) is the mean number of bosons at well n, U = 4pi~2asVeff/m de- scribes the interaction between two atoms on a single site (Veff is the effective mode volume of each site, m is the atomic mass, and as is the s-wave atomic scattering length), µn is the well chemical potential, and T is the tunneling amplitude. The "wavefunction amplitudes" ψn(t) ≡ √ Nn(t) exp(−iφn(t)) can be used as conjugate variables with respect to the Hamiltonian iH leading to a set of canonical equations i∂tψn = ∂H/∂ψ∗n; i∂tψ∗n = −∂H/∂ψn. Substituting (4) we get the DNLSE. To sim- ulate the leaking process at the two edges, we supplement the standard DNLSE with a local dissipation at the two edges of the lattice. The resulting leaking DNLSE is [10] i ∂ψn ∂τ = (χ |ψn|2 + µ˜n)ψn − 12 [ψn−1(1− δn,1) + ψn+1(1− δn,M )] − iγψn[δn,1 + δn,M ]; n = 1, · · · ,M (5) where the normalized time is defined as τ = Tt, χ = 2U/T is the rescaled nonlinearity, µ˜n = µn/T is the rescaled chemical potential and γ is the atom emission probability describing atomic losses. It is useful to de- fine the initial effective interaction per well i.e. Λ = χρ where ρ = N(τ = 0)/M is the initial average density of atoms in the OL. In our numerical experiments we have used initial conditions with randomly distributed phases, and an almost constant amplitude with only small ran- dom fluctuations across the OL. We normalized the wave functions such that N(τ = 0) = 1. The initial states are first thermalized during a conservative (i.e. γ = 0) transient period of, typically, τ = 500. The dissipation at the lattice boundaries is switched on only after this transient is completed, leading to a progressive loss of atoms. We study the decay and the statistical properties of N(τ) as a function of the parameter Λ. The results reported in this Letter correspond to µ˜n = 0 and dissi- pation rate γ = 0.2 which is within the experimentally accessible range [10]. Nevertheless we have checked that we get the same qualitative behavior for other values of γ [14]. [1] B. P. Anderson, and M. A. Kasevich, Macroscopic quan- tum interference from atomic tunnel arrays, Science 282, 1686 (1998). [2] I. Bloch, Ultracold quantum gases in optical lattices, Na- ture Phys. 1, 23-30 (2005). [3] C. Orzel, A. K. Tuchman, M. L. Fenselau, M. Yasuda, M. A. Kasevich, Squeezed states in a Bose-Einstein Con- densate, Science 291, 2386 (2001). [4] M. Greiner, O. Mandel, T. Esslinger, T. W. Hansch, I. Bloch, Quantum phase transition from a superfluid to a Mott insulator in a gas of ultracold atoms, Nature 415, 39 (2002). [5] F. S. Cataliotti, S. Burger, C. Fort, P. Maddaloni, F. Mi- nardi, A. Trombettoni, A. Smerzi, M. Inguscio, Joseph- son junction arrays with Bose-Einstein condensates, Sci- ence 293 ,843 (2001) [6] J. E. Lye, L. Fallani, M. Modugno, D. S. Wiersma, C. Fort, M. Inguscio, Bose-Einstein Condensate in a Ran- dom Potential, Phys. Rev. Lett. 95, 070401 (2005); C. Fort, L. Fallani, V. Guarrena, J. E. Lye, M. Modugno, D. S. Wiersma, M. Inguscio, Effect of Optical Disorder and Single Defects on the Expansion of a Bose-Einstein Con- densate in a One-Dimensional Waveguide, Phys. Rev. Lett. 95, 170410 (2005). [7] T. Schulte, S. Drenkelforth, J. Kruse, W. Ertmer, J. Arlt, K. Sacha, J. Zakrzewski, M. Lewenstein, Routes To- wards Anderson-like Localization of Bose-Einstein Con- densates in Disordered Optical Lattices, Phys. Rev. Lett. 95, 170411 (2005). [8] D. Clement, A. F. Varon, M. Hugbart, J. A. Retter, P. Bouyer, L. Sanchez-Palencia, D. M. Gangardt, G. V. Shlyapnikov, A. Aspect, Suppression of Transport of an Interacting Elongated Bose- Einstein Condensate in a Random Potential, Phys. Rev. Lett. 95, 170409 (2005). [9] B. Damski, J. Zakrzewski, L. Santos, P. Zoller, M. Lewen- stein, Atomic Bose and Anderson Glasses in Optical Lat- tices, Phys. Rev. Lett. 91, 080403 (2003). [10] R. Livi, R. Franzosi, G-L Oppo, Self-Localization of Bose- Einstein Condensates in Optical Lattices via Boundary Dissipation, Phys. Rev. Lett. 97, 060401 (2006); R. Fran- zosi, R. Livi, G-L. Oppo, Probing the dynamics of Bose- Einstein Condensates via boundary dissipation, J. Phys. B: At. Mol. Opt. Phys. 40, 1195 (2007). [11] I. Bloch, T. W. Hänsch, T. Esslinger, Phys. Rev. Lett. 82, 3008 (1999); A. Öttl et al. , Phys. Rev. Lett. 95, 090404 (2005). [12] D. N. Christodoulides and R. I. Joseph, Opt. Lett. 13, 794 (1988); Phys. Rev. Lett. 62, 1746 (1989); D. Mandelik et al., ibid. 90, 053902 (2003); T. Kottos, M. Weiss, Phys. Rev. Lett. 93, 190604 (2004). [13] A. Trombettoni and A. Smerzi, Phys. Rev. Lett. 86, 2353 (2001). [14] Gim Seng Ng, Holger Hennig, Ragnar Fleischmann, Tsampikos Kottos and Theo Geisel (in preparation); G. 4 S. Ng, Singnatures of Phase Transition in Wave Dynam- ics of Complex Systems, Honor Thesis, Wesleyan Univer- sity (2008). [15] B. I. Swanson, J. A. Brozik, S. P. Love, G. F. Strouse, A. P. Shreve, A. R. Bishop, W.-Z. Wang, and M. I. Salkola, Observation of Intrinsically Localized Modes in a Dis- crete Low-Dimensional Material, Phys. Rev. Lett. 82, 3288 (1999); K Kladko, J Malek and A R Bishop, In- trinsic localized modes in the charge-transfer solid PtCl, J. Phys.: Condens. Matter 11, L415 (1999). [16] U. T. Schwarz, L. Q. English, and A. J. Sievers, Ex- perimental Generation and Observation of Intrinsic Lo- calized Spin Wave Modes in an Antiferromagnet, Phys. Rev. Lett. 83, 223 (1999). [17] E. TrÃas, J. J. Mazo, and T. P. Orlando, Discrete Breathers in Nonlinear Lattices: Experimental Detec- tion in a Josephson Array, Phys. Rev. Lett. 84, 741 (2000); L. M. FlorÃa, J. L. Marín, P. J. Martínez, F. Falo and S. Aubry, Intrinsic localisation in the dynam- ics of a Josephson-junction ladder, Europhys. Lett. 36, 539 (1996); A. V. Ustinov, Imaging of discrete breathers, Chaos 13, 716 (2003). [18] M. Sato, B. E. Hubbard, A. J. Sievers, B. Ilic, D. A. Czaplewski, and H. G. Craighead, Observation of Locked Intrinsic Localized Vibrational Modes in a Micromechan- ical Oscillator Array, Phys. Rev. Lett. 90, 044102 (2003). [19] H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, Discrete Spatial Optical Soli- tons in Waveguide Arrays, Phys. Rev. Lett. 81, 3383 (1998); R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, Dynamics of Discrete Soli- tons in Optical Waveguide Arrays, Phys. Rev. Lett. 83, 2726 (1999); D. N. Christodoulides and R. I. Joseph, Vec- tor solitons in birefringent nonlinear dispersive media, Opt. Lett. 13, 53 (1988); Yuri S. Kivshar, Self-localization in arrays of defocusing waveguides, Opt. Lett. 18, 1147 (1993). [20] Jason W. Fleischer, Mordechai Segev, Nikolaos K. Efremidis, and Demetrios N. Christodoulides, Observa- tion of two-dimensional discrete solitons in optically in- duced nonlinear photonic lattices, Nature 422, 147 (2003); Serge F. Mingaleev and Yuri S. Kivshar, Self-Trapping and Stable Localized Modes in Nonlinear Photonic Crys- tals, Phys. Rev. Lett. 86, 5474 (2001). [21] S. Flach and C. R. Willis, Discrete breathers, Physics Reports 295, 181 (1998); S. Flach, C. R. Willis, E. Ol- brich, Integrability and localized excitations in nonlinear discrete systems, Phys. Rev. E 49, 836 (1994). [22] A. R. Bishop, G. Kalosakas, K. O. Rasmussen, Localiza- tion in physical systems described by discrete nonlinear Schroedinger-type equations, Chaos 13, 588 (2003). [23] P. G. Kevrekidis, K. Ø. Rasmussen, and A. R. Bishop, The Discrete nonlinear Schroedinger Equation: A survey of recent results, Int. J. Mod. Phys. B 15, 28