Controlling Chaotic Behavior in a Bose Einstein Condensate with Linear Feedback

Commun. Theor. Phys. (Beijing, China) 50 (2008) pp. 215 219 c Chinese Physical Society Vol. 50, No. 1, July 15, 2008 Controlling Chaotic Behavior in a Bose Einstein Condensate with Linear Feedback WANG Zhi-Xia, 1,2, ZHANG Xi-He, 1 and SHEN Ke 1 1 Department of Physics, Changchun University of Science and Technology, Changchun 130022, China 2 Aviation University of Air Force, Changchun 130022, China (Received September 12, 2007; Revised November 13, 2007) Abstract The spatial structure of a Bose Einstein condensate (BEC) loaded into an optical lattice potential is investigated and the spatially chaotic distributions of the condensates are revealed. A method of chaos control with linear feedback is presented in this paper. By using the method, we propose a scheme of controlling chaotic behavior in a BEC with atomic mirrors. The results of the computer simulation show that controlling the chaos into the stable states could be realized by adjusting the coefficient of feedback only if the maximum Lyapunov exponent of the system is negative. PACS numbers: 42.65.Sf, 37.10.Jk, 42.50.Md Key words: controlling chaos, BEC, linear feedback, Lyapunov exponent 1 Introduction Eighty years after its prediction, BEC has been observed in trapped gases of rubidium, sodium, and lithium. [1] The mean-field theory (GP equation) has been quite successful in reproducing quantitatively many experimental observations. [2] The realization of BEC in dilute alkali vapors has opened the field of weakly-interacting degenerate Bose gas. Subsequential experimental and theoretical progress has been made on the study of the properties of this new state of matter. Several remarkable phenomena, which strongly resemble well-known effects in nonlinear optics, have been observed in BEC, such as four-wave mixing, vortices, dark, bright solitons and chaos. [3 11] In realistic experimental setting, external electromagnetic field is used to produce, trap, and manipulate BEC. In early experiments, only harmonic potential was employed, but a wide variety of potentials can now be constructed experimentally. Among the most frequently studied both experimentally and theoretically, are periodic optical lattice potentials. The optical lattice is created as standing-wave interference pattern mutually coherent laser beams. With each lattice site occupied by one mass of alkali atoms in its ground state, BEC in optical lattices shows a number of potential applications, such as atomic interferometer, registers for quantum computers, atom laser, quantum information processing on the nanometer scale and others. Optical lattice is, therefore, of particular interest from the perspective of both fundamental quantum physics and its connection to applications. [8] Numerous experimental studies have confirmed the general validity of the timedependent nonlinear Schrödinger equation, also called GP equation, used to calculate the ground state and excitations of various BEC of trapped alkali atoms. The dynamics of the system is described by a Schrödinger equation combining with a nonlinear term, which represents the many-body interactions, in the mean-field approximation. This nonlinearity makes it possible to bring chaos into the quantum system. The existence of BEC chaos has been proved and the chaotic properties have also been extensively researched in many previous works. Naturally, chaos, which plays a role in the regularity of the system, will cause instability of the condensate wave function. The study of chaos in nonlinear deterministic system has been developed for many years. Besides developing the basic questions about the mechanisms and the predictions of chaos, however, the ability to control it to a regular state is also an important subject for the relevant scientists. For the purpose of applications control of chaos is anticipated in practical investigations. Only a few letters have investigated controlling chaos in a BEC. One is controlling chaos in a weakly coupled array of BEC by using the Ott Grebogi Yorker (OGY) scheme. [12] Chaos control has always been a widely attractive field since the pioneering work of OGY in 1990. [13] Controlling chaos can be separated into two categories: feedback control (active control) and nonfeedback control (passive control). The general method for feedback control is to push a system state onto a stable manifold of a target orbit that is to say, stabilizing the unstable target orbits embedded within a chaotic attractor. Chen proposed linear feedback in 1993. [14,15] Recently, much interest was focused on the chaotic behavior of a BEC loaded into an optical lattice potential. The main purpose of the present paper is to control the chaos into the stable states in BEC with atomic mirrors by using linear feedback. [16] For different coefficients of feedback, the evolutional trajectories of the atomic number density fall onto different regular E-mail: wzx2007111@126.com

216 WANG Zhi-Xia, ZHANG Xi-He, and SHEN Ke Vol. 50 attractors. We can force the system to the stable periodic orbit. 2 Controlling Chaos in BEC System with Linear Feedback The BEC system considered here is created in a harmonically trapped potential and then is loaded into a moving optical lattice. The 3D combined potential therefore is given by V (x, y, z, t 1 ) = V 1 cos 2 (kξ) + m(ω 2 xx 2 + ω 2 yy 2 + ω z z 2 )/2, where the second term is the harmonically magnetic potential with m being the atomic mass and ω x, ω y, ω z the trap frequencies. The periodic potential is a moving optical lattice with the space-time variable ξ = x + δt 1 /2k, where δ is the frequency difference between the two counterpropagating laser beams and k the laser wave vector, which fixes the velocity of the traveling lattice as V L = δ/2k. When the BEC is formed in the region near the center of the magnetic trap, the magnetic potential is much weaker than the lattice one and can be neglected. We find that in the region k x 2 + (y 2 /2) + (z 2 /4) 100π the harmonic potential is much less than the lattice potential. Therefore, the 1D optical potential plays the main role for the system and the quasi-1d approximation is valid in this region. On the other hand, for a time-dependent lattice, the damping effect should be considered. The damping effect caused by the incoherent exchange of normal atoms and the finite temperature effect has been analyzed in detail for the two-junction linking of two BECs. For the system considered here, it is similar to the case of the linear junction linking of many BECs. Thus, a damping effect caused by similar elements or other factors may also exist. With these considerations, the system is governed by the following quasi-1d GP equation: i (1 iλ) ψ = 2 2 ψ t 1 2m x 2 + g 0 ψ 2 ψ + v 1 cos 2 (kξ)ψ, (1) where m is the atomic mass and g 0 = 4π 2 a/m denotes the interatomic interaction with a being the s-wave scattering length. The case a > 0 represents a repulsive interatomic interaction, and a < 0 implies the attractive case. Parameter ψ is the macroscopic quantum wave function. The term proportional to γ represents the damping effect. We focus our interest on only the traveling wave solution of this equation and write Eq. (1) in the form ψ = ϕ(ξ) e i(αx+βt 1), (2) such that the matter wave is a Bloch-like wave. Here, α and β are two undetermined real constants. According to the definition of the space-time variable ξ = x+v L t 1 in the former, the traveling wave ϕ(ξ) moves with the same velocity as the optical lattice. Inserting Eq. (2) into Eq. (1), we can easily turn the partial differential equation (1) into an ordinary differential one: 2 d 2 ϕ ( 2 2m dξ 2 + i α ) dϕ ( m + v L i γv L dξ β + 2 α 2 ) 2m i βγ ϕ g 0 ϕ 2 ϕ = v 1 cos 2 (kξ)ϕ. (3) For simplicity, using the dimensionless variables and parameters t = kξ, v = 2mv L k, α 1 = α k, I 0 = v 1 E r. β 1 = β E r, We let ϕ = R(t) e iθ(t) and dθ/dt = β 1 /v = (v/2)+ α 1. We have Eq. (3) in the form dy 1 dt = dr dt = y 2, dy 2 dt = d2 R dt 2 = 1 4 v2 y 1 + gy 3 1 + I 0 cos 2 (t)y 1 γvy 2, (4) where I 0 is optical intensity, v = 2mv L / k. The square of the amplitude R is just the particle number density because R = ϕ = ψ, and θ is the phase of ϕ. [5] According to the general theory of the Duffing equation, underlying Eq. (4) has a monoclinic solution only when the coefficients of the linear (R) and nonlinear (R 3 ) terms on the left-hand side of Eq. (4) have opposite signs. Therefore, in order to study the chaos for the negative R term we must consider the case of attractive atom-atom interactions, i.e., g < 0; equation (4) is just the parametrically driven Duffing equation with a damping term. We solve Eq. (4) numerically by using the fourth Runge Kutta (R-K) algorithms. In order to avoid transient chaos, the y 1 and y 2 in the initial 20 000 steps are eliminated. The initial conditions are y 1 = 1.0, y 2 = 0.0, t = 0.0. The parameters in Eq. (4) are listed as follows: I 0 = 5.5, γ = 0.05, v = 2.03, g = 0.75. Figure 1(a) shows the strange attractor projected on to the y 1 -y 2 plane; however, we cannot tell whether this attractor is chaotic. We calculate the maximum Lyapunov exponent of the BEC system by using the algorithms presented by Wolf et al. [11,15] The maximum Lyapunov exponent of the BEC system is LY max = 0.4107. The system lies in chaotic state, as there exists one positive Lyapunov exponent. Figure 1(b) shows the time series of y 1, and one can find that the value seems random, but it is different from those noise signals without rules and seems to change following some regularity.

No. 1 Controlling Chaotic Behavior in a Bose Einstein Condensate with Linear Feedback 217 made. [16] In this scheme the output atoms from BEC system are returned to the initial BEC system in proper intensity through atomic mirror. P is coefficient of feedback. BEC is not controlled when P is 0. 3 Numerical Simulation In order to see clearly the dependence of the chaotic regions about the system coefficient of feedback, starting from Eq. (5), we plot the maximum Lyapunov exponent LY max versus the coefficient of feedback p in Fig. 2. Here the parameters are taken as γ = 0.05, v = 2.03, g = 0.75, and I 0 = 5.5. The middle point-drawing line stands for the value of zero. Figure 3 shows the detailed changes of Fig. 2. Fig. 1 The chaotic attractor and the corresponding time series with I 0 = 5.5, γ = 0.05, v = 2.03, and g = 0.75. In order to control the chaos in a BEC loaded into a moving optical lattice potential, we adjust intensity of cold atom with atomic mirrors and can obtain Eq. (5), dy 1 dη = dr dη = y 2, dy 2 dη = d2 R dη 2 = 1 4 v2 y 1 + gy 3 1 + I 0 cos 2 (η)y 1 γvy 2 py 2. (5) Atomic mirror is one of the important atom-optical elements. Evanescent-light-wave atomic mirror, semi-gaussian-beam atomic mirror, periodically-magnetized videotape mirror, periodically-arranged permanent-magnets mirror and current-carrying-wire mirror have been Fig. 2 The maximum Lyapunov LY max as a function of the coefficient of feedback p with γ = 0.05, v = 2.03, g = 0.75, and I 0 = 5.5. We find that in many ranges, for example p 0.48, p 0.5, 0.53 < p < 0.72, and p > 0.952, the maximum Lyapunov exponent is negative. If the coefficient of feedback p takes the value in these ranges, the BEC would be in periodic state. The BEC would be in different periodic state when the coefficient of feedback p takes values as 0.48, 0.5, 0.55, 0.954, and 0.966. Fig. 3 The detail changes of Fig. 2. We solve Eqs. (5) numerically by using the fourth R-K algorithms. Figure 4 shows the attractor projected onto the y 1 -y 2 plane and the time series of y 1. In order to avoid transient chaos, the y 1 and y 2 of 30 000 steps are solved. The y 1 and y 2 of the initial 20 000 steps are eliminated. The last 10 000 steps of y 1 and y 2 are retained. The initial

218 WANG Zhi-Xia, ZHANG Xi-He, and SHEN Ke Vol. 50 conditions are y 1 = 0.1, y 2 = 0.0, and t = 0.0. The parameters in Fig. 4 are listed as follows: I 0 = 5.5, γ = 0.05, v = 2.03, g = 0.75. One can find that figures 4(a), 4(c), 4(e), 4(g), and 4(i) are in periods 6, 2, 1, 2, 1 respectively for (a) and (b) with p = 0.48, (c) and (d) with p = 0.5, (e) and (f) with p = 0.55, (g) and (h) with p = 0.954, (i) and (j) with p = 0.966. Figure 4(b) is the power spectrum according to Fig. 4(a). Figure 4(d) is the time series according to Fig. 4(c). Figure 4(f) is the power spectrum according to Fig. 4(e). Figure 4(h) is the time series according to Fig. 4(g) and figure 4(j) is the time series according to Fig. 4(i). Fig. 4 The various periodic attractors, the corresponding time series and the power spectra at different coefficients of feedback p with γ = 0.05, v = 2.03, g = 0.75, and I 0 = 5.5. (a)(b) p=0.48; (c)(d) p =0.5; (e)(f) p = 0.55; (g)(h) p = 0.954; (i)(j) p = 0.966.

No. 1 Controlling Chaotic Behavior in a Bose Einstein Condensate with Linear Feedback 219 4 Conclusion In summary, we have investigated the chaotic features in the spatial distributions of the BEC. The chaotic features in the spatial distributions of the BEC for the stationary state were revealed. This kind of chaotic property maybe has a destructive role in the formation of the matter wave and is also not expected in the application of the BEC. We present a method of control of chaos via modulating the coefficient of feedback p. The method is linear feedback. In this scheme the output atoms from BEC system are returned to the initial BEC system in proper intensity through atomic mirror. Numerical simulation shows that the period is different for different modulation coefficients of feedback p. The two-mode theory, which is a simple model, already provides some interesting properties of the BEC system. We further can consider that there are three-body interaction terms in the BEC system and they will result in some new interesting results. Chaos is associated with quantum entanglement and quantum error correcting, which are both the fundamental subjects in quantum computations. Thus, it is valuable to apply or control the chaos in the BEC systems. References [1] M.II. Anderson, J.R. Ensher, M.R. Mauhews, C.E. Wieman, and E.A. Comell, Science 269 (1995) 198; K.B. Davis, M.O. Mewes, M.R. Andrews, N.J. van Dan Druten, D.S. Durfee, D.M. Kurn, and W. Keuerle, Phys. Rev. Lett. 75 (1995) 3969; C.C. Bradley, C.A. Sackeu, J.J. Tolleu, and R.G. Hulet, Phys. Rev. Lett. 75 (1995) 1687. [2] F. Dalfovo, S. Giorgmi, L.P. Pnacvkii, and S. Stringart, Rev. Mod. Phys. 71 (1999) 463, and references therein. [3] V.S. Filho, A. Gammal, T. Frederio, and L. Tomio, Phys. Rev. A 62 (2000) 033605; H.R. Brand and R.J. Deissler, Phys. Rev. E 58 (1998) R4064. [4] W. Hai, C. Lee, G. Chong, and L. Shi, Phys. Rev. E 66 (2002) 026202; Q. Xie and W. Hai, Chaos 13 (2003) 801; C. Lee, W. Hai, L. Shi, X. Zhu, and K. Gao, Phys. Rev. A 64 (2001) 053604. [5] G. Chong, W. Hai, and Q. Xie, Chaos 14 (2004) 217; Phys. Rev. E 70 (2004) 036213. [6] Yu. Kagan, E.L. Surkov, and G.V. Shlyapnikov, Phys. Rev. A 55 (1997) R18; J. Denschlag, J.E. Simsarian, D.L. Feder, and Caarles W. Clark, Science 287 (2000) 97. [7] P. Coullet and N. Vandenberghe, Phys. Rev. E 64 (2001) 025202; E.K. Strecker, B.G. Partridge, and Andrew G. Truscott, Nature (London) 417 (2002) 150. [8] R. Franzosi and V. Penna, Phys. Rev. E 67 (2003) 046227; Lacopo Carusotto, Davide Embriaco, and Giuseppe C.Lakocca, Phys. Rev. A 65 (2002) 053611; Yang-Shan Cheng, Hong Li, and De-Jing Yin, Chin. Opt. Lett. 3 (2005) 305. [9] Q. Thommen, J.C. Garreau, and U. Zehnle, Phys. Rev. Lett. 91 (2003) 210405; ZHANG Xiao-Fei, XIE Chong- Wei, and WANG Shi-En, Commun. Theor. Phys. (Beijing, China) 47 (2007) 1063. [10] H. Saito and M. Ueda, Phys. Rev. Lett. 86 (2001) 1406; XU Zhi-Jun and CAI Ping-Gen, Commun. Theor. Phys. (Beijing, Cina) 47 (2007) 1053. [11] V.I. Kuvshinov, A.V. Kuzmin, and R.G. Shulyakovsky, Phys. Rev. E 67 (2003) 015201; A. Wolf, J.B. Swift, H.L. Swinney, et al., Physica D 16 (1985) 285. [12] Gui-Shu Chong, Wen-Hua Hai, and Qiong-Tao Xie, Phys. Rev. E 71 (2005) 016202. [13] S. Sato, M. Sano, and Y. Sawada, Phys. Rev. A 28 (1993) 1654; E. Ott, C. Grebogi, and J.A. Yorke, Phys. Rev. Lett. 64 (1990) 1196. [14] G. Chen, et al., J. Circuits Syst. Comput. 3 (1993) 139. [15] G. Chen, in Proc. IEEE Inst. Symp. on Circuits and Systems, Chicago, Vol. 1 (1993) p. 2601. [16] Yong Xia and Jian-Ping Yin, J. Atomic Mol. Phys. 22 (2005) 385.