Phase separation of non-topological states in trapped two-component Bose-Einstein condensates Ricardo Carretero http://www.rohan.sdsu.edu/ ∼ rcarrete Nonlinear Dynamical Systems Group http://nlds.sdsu.edu/ Computational Science Research Center Department of Mathematics and Statistics San Diego State University Nonlinear Dynamical Systems – SDSU LENCOS, Sevilla, Spain, 14-17 Jul 2009 – p.1/22
Collaborators/Links (I) Nonlinear Dynamical Systems @ SDSU: http://nlds.sdsu.edu/ Peter Blomgren (Numerical PDEs, image processing) Ricardo Carretero (App. math., nonlinear lattices and waves) Joe Mahaffy (Mathematical biology, delay differential equations) Antonio Palacios (Applied mathematics, bifurcations, symmetries) Diana Verzi (Mathematical biology, Mathematical Physiology) Nonlinear Dynamical Systems – SDSU LENCOS, Sevilla, Spain, 14-17 Jul 2009 – p.2/22
Collaborators/Links (I) Nonlinear Dynamical Systems @ SDSU: http://nlds.sdsu.edu/ Peter Blomgren (Numerical PDEs, image processing) Ricardo Carretero (App. math., nonlinear lattices and waves) Joe Mahaffy (Mathematical biology, delay differential equations) Antonio Palacios (Applied mathematics, bifurcations, symmetries) Diana Verzi (Mathematical biology, Mathematical Physiology) Research Students involved in BECs/nonlinear waves Rafael Navarro, Ron Caplan, Eunsil Baik (PhD, Comp. Sci.). Carlos Prieto, Max Rietmann, Suchitra Jagdish (MS, Dyn. Syst.). Recent departures: Manjun Ma (2008, Postdoc), John Everts (2008), Mike Davis (2007), Chris Chong (2006) (MS, Dyn. Syst.). Nonlinear Dynamical Systems – SDSU LENCOS, Sevilla, Spain, 14-17 Jul 2009 – p.2/22
Collaborators in Nonlinear Waves/Lattices, BECs (II) Solitons, Vortices and Vortex Lattices Augusto Rodrigues (Porto) Panos Kevrekidis (UMass) Vladimir Konotop (Lisboa) D. Frantzeskakis (Athens) Yuri Kivshar (Camberra) Boris Malomed (Tel Aviv) Bernard Deconinck (UoW) Faustino Palmero (Sevilla) Enam Hoq (WNEC) Jesus Cuevas (Sevilla) Mason Porter (Caltech) Nathan Kutz (UoW) Jared Bronski (UI-UC) Yuliy Bludov (Lisboa) Peter Engels (WSU) Yannis Kevrekidis (Princeton) Dimitri Maroudas (UMass) David Hall (Amherst Coll.) George Theocharis (UMass) Brian Anderson (UoA) Hector Nistazakis (Athens) W. Królikowski (CUDOS/ANU) Alan Bishop (LANL) Chiara Daraio (CalTech) Ricardo Chacón (Badajoz) Hadi Susanto (Nottingham) Yaroslav Kartashov (ICFO) Todd Kapitula (UNM) Keith Promislow (SFU/MSU). Lluis Torner (ICFO) Lincoln Carr (Col. Sch. Mines) etc, ... Nonlinear Dynamical Systems – SDSU LENCOS, Sevilla, Spain, 14-17 Jul 2009 – p.3/22
New Book — BECs: Theory and Experiment. Nonlinear Dynamical Systems – SDSU LENCOS, Sevilla, Spain, 14-17 Jul 2009 – p.4/22
Goals/Intro Interactions between two atomic species in a binary BEC Immiscibility conditions for non-topological states Statics and dynamics of mixed and separated states Understand bifurcation scenario of higher-order mixed states Nonlinear Dynamical Systems – SDSU LENCOS, Sevilla, Spain, 14-17 Jul 2009 – p.5/22
Goals/Intro Interactions between two atomic species in a binary BEC Immiscibility conditions for non-topological states Statics and dynamics of mixed and separated states Understand bifurcation scenario of higher-order mixed states 2 hyperfine states of the same atom (cf. 87 Rb in D. Hall’s group), After adim and dimensionality reduction: ∂x 2 + Ω 2 ∂ 2 i∂u 1 � − 1 � 2 x 2 + | u 1 | 2 + g | u 2 | 2 = u 1 , ∂t 2 ∂x 2 + Ω 2 ∂ 2 � � i∂u 2 − 1 2 x 2 + | u 2 | 2 + g | u 1 | 2 = u 2 , ∂t 2 Linear vs nonlinear coupling (Boris). Nonlinear Dynamical Systems – SDSU LENCOS, Sevilla, Spain, 14-17 Jul 2009 – p.5/22
Method: Variational Approximation (VA) � ∞ 2C Lagrangian: L = −∞ ( L 1 + L 2 + L 12 + L 21 ) dx, ∂ u ∗ E j + i � ∂u j � j L j = u j − u ∗ , j 2 ∂t ∂t 2 � � 1 ∂u j + V ( x ) | u j | 2 + 1 2 | u j | 4 , � � E j = � � 2 ∂x � � L 21 = 1 2 g | u 1 | 2 | u 2 | 2 , L 12 = Nonlinear Dynamical Systems – SDSU LENCOS, Sevilla, Spain, 14-17 Jul 2009 – p.6/22
Method: Variational Approximation (VA) � ∞ 2C Lagrangian: L = −∞ ( L 1 + L 2 + L 12 + L 21 ) dx, ∂ u ∗ E j + i � ∂u j � j L j = u j − u ∗ , j 2 ∂t ∂t 2 � � 1 ∂u j + V ( x ) | u j | 2 + 1 2 | u j | 4 , � � E j = � � 2 ∂x � � L 21 = 1 2 g | u 1 | 2 | u 2 | 2 , L 12 = Gaussian ansatz: Ae − ( x − B )2 2 W 2 e i ( C + Dx + Ex 2 ) , u 1 ( x, t ) = Ae − ( x + B )2 2 W 2 e i ( C − Dx + Ex 2 ) . u 2 ( x, t ) = Time depend. params: A ( t ) , B ( t ) , C ( t ) , D ( t ) , E ( t ) , W ( t ) . Nonlinear Dynamical Systems – SDSU LENCOS, Sevilla, Spain, 14-17 Jul 2009 – p.6/22
GPE → ODEs Lagrangian evaluated @ ansatz + Euler-Lagrange eqs: dA = − AE, dt dB = D + 2 BE, dt √ 2 W 2 − D 2 B 2 2 A 2 dC 1 8 W 2 (2 B 2 − 5 W 2 ) + = 2 − 2 W 2 + dt √ 2 A 2 g 8 W 2 e − B 2 2 W 2 (8 B 4 + 2 B 2 W 2 + 5 W 4 ) , √ √ 2 A 2 Bg 2 A 2 b dD − B e − B 2 2 W 2 (4 B 2 + W 2 ) − = W 4 − 2 DE, 2 W 4 2 W 2 dt √ √ 2 A 2 g 2 A 2 2 W 4 − 2 E 2 − Ω 2 dE 1 4 W 4 e − B 2 2 W 2 ( − 4 B 2 + W 2 ) + = 4 W 2 + 2 , dt dW = 2 EW. dt Nonlinear Dynamical Systems – SDSU LENCOS, Sevilla, Spain, 14-17 Jul 2009 – p.7/22
Statics: steady states: mixed vs separated B ∗ = 0 , √ � � 15Ω 2 + 4 µ 2 � A 2 = 2 2 8 µ − / [15(1 + g )] , ∗ � � 15Ω 2 + 4 µ 2 � W 2 / [5Ω 2 ] . = 2 µ + ∗ PDE (a) ODE 0.6 0.5 u 1 and u 2 0.4 0.3 0.2 0.1 −4 −2 0 2 4 x Nonlinear Dynamical Systems – SDSU LENCOS, Sevilla, Spain, 14 -17 Jul 2009 – p.8/22
Statics: steady states: mixed vs separated √ B 2 2 A 2 ∗ g Ω 2 − ∗ − 2 W 2 e = 0 , ∗ W 2 ∗ � √ � µ − 1 / (2 W 2 ∗ ) − 5 W 2 2 A 2 ∗ g + Ω 2 W 2 / 8 = 0 , ∗ ∗ µ + 3 / (4 W 2 ∗ ) − 5Ω 2 � W 2 ∗ + 2 B 2 � / 4 = 0 . ∗ PDE (b) ODE 0.5 0.4 u 1 and u 2 0.3 0.2 0.1 −4 −2 0 2 4 x Nonlinear Dynamical Systems – SDSU LENCOS, Sevilla, Spain, 14 -17 Jul 2009 – p.9/22
Bifurcation of steady states: PDE vs ODE C 1 Unstable (ODE) 0.8 Stable (ODE) Unstable (PDE) B 0.6 Stable (PDE) 0.4 0.2 D A 0 B −0.2 −0.4 −0.6 B −0.8 −1 (a) C 1 2 3 4 5 6 g Nonlinear Dynamical Systems – SDSU LENCOS, Sevilla, Spain, 14 -17 Jul 2009 – p.10/22
Phase separation � � � � 15Ω 2 + 4 µ 2 15Ω 2 + 4 µ 2 � � Using pt A : g cr = 6 µ + 3 / 26 µ − 7 . 7 Supercritical Pt. D (PDE) Subcritical Pt. A (ODE) 6 Saddle−Node Pt. B (ODE) 5 g 4 Stable separated state 3 2 Stable mixed state 1 0.1 0.3 0.5 0.7 Ω Nonlinear Dynamical Systems – SDSU LENCOS, Sevilla, Spain, 14 -17 Jul 2009 – p.11/22
Dynamics: oscillations trough each other 0.9 (a) 0.8 A 0.7 0.6 1 (b) B 0 −1 (c) 1 C+ µ t 0.5 0 0.5 (d) 0 D −0.5 0.2 (e) E 0 −0.2 (f) 2 W 1.5 1 2 4 6 8 10 12 14 16 18 20 t Nonlinear Dynamical Systems – SDSU LENCOS, Sevilla, Spain, 14 -17 Jul 2009 – p.12/22
Dynamics: oscillations about fixed pts: separated osc. 0.7 (a) 0.6 A 0.5 (b) 1 B 0 −1 (c) −0.5 −1 C+ µ t −1.5 −2 −2.5 (d) 0.5 0 D −0.5 (e) 0.2 E 0 −0.2 1.4 (f) 1.2 W 1 0.8 2 4 6 8 10 12 14 16 18 20 t Nonlinear Dynamical Systems – SDSU LENCOS, Sevilla, Spain, 14 -17 Jul 2009 – p.13/22
Understanding the phase separation → U eff Newtonian reduction: 4.5 g=20 A ( t ) ≈ A ∗ and W ( t ) ≈ W ∗ 4 Newton oscillations 3.5 d 2 B dt 2 = − dU eff ( B ) 3 , dB 2.5 U eff inside the effective potential 2 √ U eff = Ω 2 2 A 2 g=6 − B 2 ∗ g 2 B 2 + 2 W 2 ∗ . 1.5 e 2 1 g=2.6 which becomes double well 0.5 for large enough g ( g > g cr ) g=0 0 −5 0 5 B Nonlinear Dynamical Systems – SDSU LENCOS, Sevilla, Spain, 14 -17 Jul 2009 – p.14/22
Bifurcation of steady states 1 0.8 0.6 0.4 0.2 ∆ h 0 A B C D E −0.2 −0.4 −0.6 −0.8 −1 0.5 1 1.5 2 2.5 3 3.5 4 4.5 g Nonlinear Dynamical Systems – SDSU LENCOS, Sevilla, Spain, 14 -17 Jul 2009 – p.15/22
Dynamics of unstable mixed state (for g > g cr ) (e) t=0 (b) t= 0 0.4 0.4 t=10000 |u 1,2 (x)| 2 |u 1,2 (x)| 2 t=40 0.2 0.2 0 0 −5 0 5 −5 0 5 x x (c) 0.4 t= 0 |u 1,2 (x)| 2 t=145 0.2 0 −5 0 5 x Nonlinear Dynamical Systems – SDSU LENCOS, Sevilla, Spain, 14 -17 Jul 2009 – p.16/22
Dynamics of (unstable) higher order states (d) t=0 (b) t= 0 0.4 0.4 |u 1,2 (x)| 2 t=15000 t=74 |u 1,2 (x)| 2 0.2 0.2 0 0 −5 0 5 −5 0 5 x x Same qualitative ‘decay’ for other states All unstable states ‘decay’ to a highly perturbed separated state. [Except the (unstable) 1 -2 hump state] Nonlinear Dynamical Systems – SDSU LENCOS, Sevilla, Spain, 14-17 Jul 2009 – p.17/22
Bifurcation scenario for asymmetric states ( µ 1 � = µ 2 ) A B C D E F S N T G L T M F G E C,K,R 0 Var(u 1 )−Var(u 2 ) D S H B A H R I J I −0.1 Q 0 P Q J 0.6 1 1.4 1.8 µ 1 Nonlinear Dynamical Systems – SDSU LENCOS, Sevilla, Spain, 14 -17 Jul 2009 – p.18/22 P O N M L K
Outlook & to do’s (I) Problem motivated by the experiments from David Hall in binary 87 Rb. [cf. our work in PRL 99 , 190402 (2007)]. Nonlinear Dynamical Systems – SDSU LENCOS, Sevilla, Spain, 14 -17 Jul 2009 – p.19/22
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