Conditions for order and chaos in the dynamics of a trapped Bose-Einstein condensate in coordinate and energy space

Abstract We investigate numerically conditions for order and chaos in the dynamics of an interacting Bose-Einstein condensate (BEC) confined by an external trap cut off by a hard-wall box potential. The BEC is stirred by a laser to induce excitations manifesting as irregular spatial and energy oscillations of the trapped cloud. Adding laser stirring to the external trap results in an effective time-varying trapping frequency in connection with the dynamically changing combined external+laser potential trap. The resulting dynamics are analyzed by plotting their trajectories in coordinate phase space and in energy space. The Lyapunov exponents are computed to confirm the existence of chaos in the latter space. Quantum effects and trap anharmonicity are demonstrated to generate chaos in energy space, thus confirming its presence and implicating either quantum effects or trap anharmonicity as its generator. The presence of chaos in energy space does not necessarily translate into chaos in coordinate space. In general, a dynamic trapping frequency is found to promote chaos in a trapped BEC. An apparent means to suppress chaos in a trapped BEC is achieved by increasing the characteristic scale of the external trap with respect to the condensate size. Graphical abstract

[1]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[2]  Miss A.O. Penney.  (b) , 1974, The New Yale Book of Quotations.

[3]  O. Rössler.  An equation for continuous chaos , 1976 .

[4]  James P. Crutchfield,et al.  Low-dimensional chaos in a hydrodynamic system , 1983 .

[5]  Heller,et al.  Semiclassical dynamics of chaotic motion: Unexpected long-time accuracy. , 1991, Physical review letters.

[6]  Quantum Chaotic Attractor in a Dissipative System , 1997, chao-dyn/9703012.

[7]  Y. Castin,et al.  Instability and Depletion of an Excited Bose-Einstein Condensate in a Trap , 1997 .

[8]  Reversible Formation of a Bose-Einstein Condensate , 1998, cond-mat/9805022.

[9]  J. V. José,et al.  Energy transfer of a chaotic particle in a classical oscillating potential barrier , 1998 .

[10]  C. Raman,et al.  Evidence for a Critical Velocity in a Bose-Einstein Condensed Gas , 1999 .

[11]  S. Burger,et al.  Coherent Evolution of Bouncing Bose-Einstein Condensates , 1999, cond-mat/9911309.

[12]  F. Dalfovo,et al.  Theory of Bose-Einstein condensation in trapped gases , 1998, cond-mat/9806038.

[13]  R. J. Ballagh,et al.  Coherent Dynamics of Vortex Formation in Trapped Bose-Einstein Condensates , 1999 .

[14]  C. S. Adams,et al.  Dissipation and vortex creation in Bose-Einstein condensed gases , 2000 .

[15]  Georgeot,et al.  Quantum chaos border for quantum computing , 1999, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[16]  P. Zoller,et al.  Nonlinear matter wave dynamics with a chaotic potential , 2000 .

[17]  Onofrio,et al.  Observation of superfluid flow in a bose-einstein condensed Gas , 2000, Physical review letters.

[18]  Chu,et al.  Motional dressed states in a bose-einstein condensate: superfluidity at supersonic speed , 2000, Physical review letters.

[19]  Arnaldo Gammal,et al.  Chaos in collapsing Bose-condensed gas , 2000 .

[20]  Critcal velocity of superfluid flow past large obstacles in Bose-Einstein condensates , 2000, cond-mat/0006419.

[21]  Dalibard,et al.  Vortex formation in a stirred bose-einstein condensate , 1999, Physical review letters.

[22]  R. J. Ballagh,et al.  Three-dimensional vortex dynamics in Bose-Einstein condensates , 2000 .

[23]  G Ferrari,et al.  Quasipure Bose-Einstein condensate immersed in a Fermi sea. , 2001, Physical review letters.

[24]  M D Barrett,et al.  All-optical formation of an atomic Bose-Einstein condensate. , 2001, Physical review letters.

[25]  P. Coullet,et al.  Chaotic self-trapping of a weakly irreversible double Bose condensate. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[26]  H. Smith,et al.  Bose–Einstein Condensation in Dilute Gases: Contents , 2001 .

[27]  J Dalibard,et al.  Stationary states of a rotating Bose-Einstein condensate: routes to vortex nucleation. , 2001, Physical review letters.

[28]  Carl E. Wieman,et al.  Dynamics of collapsing and exploding Bose–Einstein condensates , 2001, Nature.

[29]  Interference of a Bose-Einstein condensate in a hard-wall trap: from nonlinear Talbot effect to formation of vorticity , 1999, cond-mat/9908095.

[30]  W. Ketterle,et al.  Vortex nucleation in a stirred Bose-Einstein condensate. , 2001, Physical review letters.

[31]  M. Raizen,et al.  Optical billiards for atoms. , 2001, Physical review letters.

[32]  C. Pethick,et al.  Bose–Einstein Condensation in Dilute Gases: Contents , 2008 .

[33]  Quantum tweezer for atoms. , 2002, Physical review letters.

[34]  D. Pritchard,et al.  Transport of Bose-Einstein condensates with optical tweezers. , 2001, Physical review letters.

[35]  S. Gardiner.  (Quantum) chaos in Bose-Einstein condensates , 2002 .

[36]  Chaotic oscillation in an attractive Bose-Einstein condensate under an impulsive force , 2001, cond-mat/0112427.

[37]  D. Rychtarik,et al.  Cold-atom gas at very high densities in an optical surface microtrap , 2002, physics/0204026.

[38]  Quantum computing of quantum chaos in the kicked rotator model. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[39]  Jiannis K Pachos,et al.  Quantum computation with a one-dimensional optical lattice. , 2003, Physical review letters.

[40]  C. Adams,et al.  Soliton-sound interactions in quasi-one-dimensional Bose-Einstein condensates. , 2002, Physical review letters.

[41]  Abdelaziz Radouani,et al.  Soliton and phonon production by an oscillating obstacle in a quasi-one-dimensional trapped repulsive Bose-Einstein condensate , 2004 .

[42]  C Fort,et al.  Observation of dynamical instability for a Bose-Einstein condensate in a moving 1D optical lattice. , 2004, Physical review letters.

[43]  C. Barenghi,et al.  Parametric driving of dark solitons in atomic Bose-Einstein condensates. , 2004, Physical review letters.

[44]  Transition to instability in a kicked bose-einstein condensate. , 2003, Physical review letters.

[45]  Qiongtao Xie,et al.  Spatial chaos of trapped Bose-Einstein condensate in one-dimensional weak optical lattice potential. , 2004, Chaos.

[46]  M. Perc,et al.  Detecting chaos from a time series , 2005 .

[47]  J. Schmiedmayer,et al.  Quasicondensate growth on an atom chip , 2005, cond-mat/0509154.

[48]  L. A. Aguirre,et al.  Piecewise affine models of chaotic attractors: the Rossler and Lorenz systems. , 2006, Chaos.

[49]  Rainer Blatt,et al.  Transfer of trapped atoms between two optical tweezer potentials , 2006, quant-ph/0606018.

[50]  A. Fioretti,et al.  Optimized production of large Bose Einstein Condensates , 2006, quant-ph/0602010.

[51]  Transition to instability in a periodically kicked Bose-Einstein condensate on a ring , 2005, cond-mat/0508665.

[52]  D. R. Scherer,et al.  Vortex formation by merging of multiple trapped Bose-Einstein condensates. , 2007, Physical review letters.

[53]  P. Engels,et al.  Stationary and nonstationary fluid flow of a bose-einstein condensate through a penetrable barrier. , 2007, Physical review letters.

[54]  R. Chacón,et al.  Controlling chaos of a Bose-Einstein condensate loaded into a moving optical Fourier-synthesized lattice. , 2008, Physical review. E, Statistical, nonlinear, and soft matter physics.

[55]  E. Demiralp,et al.  Bose-Einstein condensate in a harmonic trap with an eccentric dimple potential , 2008 .

[56]  J. Belmonte-Beitia,et al.  Laser tweezers for atomic solitons , 2007, 0707.4652.

[57]  Arjendu K. Pattanayak,et al.  Nonmonotonicity in the quantum-classical transition: chaos induced by quantum effects. , 2007, Physical review letters.

[58]  C. W. Gardiner,et al.  Dynamics and statistical mechanics of ultra-cold Bose gases using c-field techniques , 2008, 0809.1487.

[59]  Matthew J. Davis,et al.  Dynamical instabilities of Bose-Einstein condensates at the band edge in one-dimensional optical lattices , 2007, 0706.2744.

[60]  Discrete chaotic states of a Bose-Einstein condensate. , 2007, Physical review. E, Statistical, nonlinear, and soft matter physics.

[61]  A. Lance,et al.  Energy distribution and cooling of a single atom in an optical tweezer , 2008, 0805.3510.

[62]  S. K. Adhikari,et al.  Fortran programs for the time-dependent Gross-Pitaevskii equation in a fully anisotropic trap , 2009, Comput. Phys. Commun..

[63]  D. Shepelyansky,et al.  Chaotic dynamics of a Bose-Einstein condensate coupled to a qubit. , 2008, Physical review. E, Statistical, nonlinear, and soft matter physics.

[64]  G. El,et al.  Stationary wave patterns generated by an impurity moving with supersonic velocity through a Bose-Einstein condensate , 2009, 0902.1317.

[65]  W. Hai,et al.  Transition probability from matter-wave soliton to chaos. , 2008, Physical review. E, Statistical, nonlinear, and soft matter physics.

[66]  Kazuki Sasaki,et al.  Bénard-von Kármán vortex street in a Bose-Einstein condensate. , 2010, Physical review letters.

[67]  Jing Cheng.  Chaotic dynamics in a periodically driven spin-1 condensate , 2010 .

[68]  Biao Wu,et al.  Atomic quantum corrals for Bose-Einstein condensates , 2008, 0812.4602.

[69]  N. Gaaloul,et al.  Bose-Einstein condensation in dark power-law laser traps , 2009, 0909.5090.

[70]  Frank Vewinger,et al.  Bose–Einstein condensation of photons in an optical microcavity , 2010, Nature.

[71]  C. Weiss,et al.  Reproducible mesoscopic superpositions of Bose-Einstein condensates and mean-field chaos , 2010, 1008.0314.

[72]  Ashton S. Bradley,et al.  Observation of vortex dipoles in an oblate Bose-Einstein condensate. , 2009, Physical review letters.

[73]  N. Katz,et al.  Parametrically excited ‘scars’ in Bose–Einstein condensates , 2009, 0903.0968.

[74]  M. Tsubota,et al.  Synergy dynamics of vortices and solitons in an atomic Bose-Einstein condensate excited by an oscillating potential , 2010, 1007.3063.

[75]  C. Adams,et al.  Dark soliton decay due to trap anharmonicity in atomic Bose-Einstein condensates , 2009, 0906.4877.

[76]  Katharina Ludwig,et al.  Wave chaos in the nonequilibrium dynamics of the Gross-Pitaevskii equation , 2011, 1101.4663.

[77]  Roger R. Sakhel,et al.  Self-interfering matter-wave patterns generated by a moving laser obstacle in a two-dimensional Bose-Einstein condensate inside a power trap cut off by box potential boundaries , 2011, 1107.0369.

[78]  M. Tsubota,et al.  Nonlinear dynamics in a trapped atomic Bose-Einstein condensate induced by an oscillating Gaussian potential , 2011, 1101.2052.

[79]  T. Tarn,et al.  Suppressing non-Markovian noises by coupling the qubit to a chaotic device , 2011, 1101.0890.

[80]  Michael C. Garrett,et al.  Growth dynamics of a Bose-Einstein condensate in a dimple trap without cooling , 2010, 1011.0296.

[81]  S. Gardiner,et al.  Coherence and instability in a driven Bose–Einstein condensate: a fully dynamical number-conserving approach , 2011, 1104.1521.

[82]  J. Dalibard,et al.  Production of sodium Bose–Einstein condensates in an optical dimple trap , 2011, 1104.1009.

[83]  Jacob F. Sherson,et al.  Quantum computation architecture using optical tweezers , 2011 .

[84]  Lian-Ao Wu,et al.  Deterministic chaos can act as a decoherence suppressor , 2011, 1101.3194.

[85]  W. Hai,et al.  Phase effects of a two-dimensional Bose–Einstein condensate , 2012 .

[86]  Lorenz S. Cederbaum,et al.  Wave chaos as signature for depletion of a Bose-Einstein condensate , 2012, 1202.5869.

[87]  Paulsamy Muruganandam,et al.  C programs for solving the time-dependent Gross-Pitaevskii equation in a fully anisotropic trap , 2012, Comput. Phys. Commun..

[88]  Roger R. Sakhel,et al.  Nonequilibrium Dynamics of a Bose-Einstein Condensate Excited by a Red Laser Inside a Power-Law Trap with Hard Walls , 2013, Journal of Low Temperature Physics.

[89]  Second-order number-conserving description of nonequilibrium dynamics in finite-temperature Bose-Einstein condensates. , 2012, 1207.2821.

[90]  Joselito M. Razal,et al.  Wet-spinning of PEDOT:PSS/Functionalized-SWNTs Composite: a Facile Route Toward Production of Strong and Highly Conducting Multifunctional Fibers , 2013, Scientific Reports.

[91]  A. Fratalocchi,et al.  Nonlinearly-enhanced energy transport in many dimensional quantum chaos , 2013, Scientific Reports.

[92]  G. Oppo,et al.  Nonlinear and chaotic dynamics of a Bose-Einstein condensate in an optical cavity , 2014 .

[93]  Stewart A. Koppell,et al.  Chaotic dynamics in a two-dimensional optical lattice. , 2014, Physical review. E, Statistical, nonlinear, and soft matter physics.

[94]  S. Wimberger.  Nonlinear Dynamics and Quantum Chaos , 2022, Graduate Texts in Physics.

[95]  S. Dutta,et al.  Kinetics of Bose-Einstein condensation in a dimple potential , 2014, 1407.2557.

[96]  Zach DeVito,et al.  Opt , 2017 .