Phase separation of non-topological states in trapped two-component - - PowerPoint PPT Presentation

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Phase separation of non-topological states in trapped two-component - - PowerPoint PPT Presentation

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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

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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

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SLIDE 2

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

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SLIDE 3

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

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SLIDE 4

Collaborators in Nonlinear Waves/Lattices, BECs (II)

Solitons, Vortices and Vortex Lattices Panos Kevrekidis (UMass)

  • D. Frantzeskakis (Athens)

Boris Malomed (Tel Aviv) Faustino Palmero (Sevilla) Jesus Cuevas (Sevilla) Mason Porter (Caltech) Yuliy Bludov (Lisboa) Peter Engels (WSU) David Hall (Amherst Coll.) Brian Anderson (UoA)

  • W. Królikowski (CUDOS/ANU)

Chiara Daraio (CalTech) Ricardo Chacón (Badajoz) Todd Kapitula (UNM) Keith Promislow (SFU/MSU). Lincoln Carr (Col. Sch. Mines) Augusto Rodrigues (Porto) Vladimir Konotop (Lisboa) Yuri Kivshar (Camberra) Bernard Deconinck (UoW) Enam Hoq (WNEC) Nathan Kutz (UoW) Jared Bronski (UI-UC) Yannis Kevrekidis (Princeton) Dimitri Maroudas (UMass) George Theocharis (UMass) Hector Nistazakis (Athens) Alan Bishop (LANL) Hadi Susanto (Nottingham) Yaroslav Kartashov (ICFO) Lluis Torner (ICFO) etc, ...

Nonlinear Dynamical Systems – SDSU

LENCOS, Sevilla, Spain, 14-17 Jul 2009 – p.3/22

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SLIDE 5

New Book — BECs: Theory and Experiment.

Nonlinear Dynamical Systems – SDSU

LENCOS, Sevilla, Spain, 14-17 Jul 2009 – p.4/22

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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

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SLIDE 7

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. 87Rb in D. Hall’s group), After adim and dimensionality reduction:

i∂u1 ∂t =

  • −1

2 ∂2 ∂x2 + Ω2 2 x2 + |u1|2 + g |u2|2

  • u1,

i∂u2 ∂t =

  • −1

2 ∂2 ∂x2 + Ω2 2 x2 + |u2|2 + g |u1|2

  • u2,

Linear vs nonlinear coupling (Boris).

Nonlinear Dynamical Systems – SDSU

LENCOS, Sevilla, Spain, 14-17 Jul 2009 – p.5/22

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Method: Variational Approximation (VA)

2C Lagrangian: L =

∞

−∞ (L1 + L2 + L12 + L21) dx,

Lj = Ej + i 2

  • uj

∂ u∗

j

∂t − u∗

j

∂uj ∂t

  • ,

Ej = 1 2

  • ∂uj

∂x

  • 2

+ V (x) |uj|2 + 1 2 |uj|4 , L12 = L21 = 1 2g |u1|2 |u2|2 ,

Nonlinear Dynamical Systems – SDSU

LENCOS, Sevilla, Spain, 14-17 Jul 2009 – p.6/22

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SLIDE 9

Method: Variational Approximation (VA)

2C Lagrangian: L =

∞

−∞ (L1 + L2 + L12 + L21) dx,

Lj = Ej + i 2

  • uj

∂ u∗

j

∂t − u∗

j

∂uj ∂t

  • ,

Ej = 1 2

  • ∂uj

∂x

  • 2

+ V (x) |uj|2 + 1 2 |uj|4 , L12 = L21 = 1 2g |u1|2 |u2|2 ,

Gaussian ansatz:

u1(x, t) = Ae− (x−B)2

2W 2 ei(C+Dx+Ex2),

u2(x, t) = Ae− (x+B)2

2W 2 ei(C−Dx+Ex2).

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

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GPE → ODEs

Lagrangian evaluated @ ansatz + Euler-Lagrange eqs:

dA dt = −AE, dB dt = D + 2BE, dC dt = B2 2W 2 − D2 2 − 1 2W 2 + √ 2A2 8W 2 (2B2 − 5W 2) + √ 2A2g 8W 2 e− B2

2W 2 (8B4 + 2B2W 2 + 5W 4),

dD dt = √ 2A2Bg 2W 4 e− B2

2W 2 (4B2 + W 2) −

√ 2A2b 2W 2 − B W 4 − 2DE, dE dt = √ 2A2g 4W 4 e− B2

2W 2 (−4B2 + W 2) +

√ 2A2 4W 2 + 1 2W 4 − 2E2 − Ω2 2 , dW dt = 2EW.

Nonlinear Dynamical Systems – SDSU

LENCOS, Sevilla, Spain, 14-17 Jul 2009 – p.7/22

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Statics: steady states: mixed vs separated

B∗ = 0, A2

∗

= 2 √ 2

  • 8µ −
  • 15Ω2 + 4µ2
  • /[15(1 + g)],

W 2

∗

=

  • 2µ +
  • 15Ω2 + 4µ2
  • /[5Ω2].

−4 −2 2 4 0.1 0.2 0.3 0.4 0.5 0.6 x u1 and u2 (a) PDE ODE

Nonlinear Dynamical Systems – SDSU

LENCOS, Sevilla, Spain, 14-17 Jul 2009 – p.8/22

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Statics: steady states: mixed vs separated

Ω2 − √ 2A2

∗g

W 2

∗

e

−

B2 ∗ 2W 2 ∗

= 0, µ − 1/(2W 2

∗ ) − 5W 2 ∗

√ 2A2

∗g + Ω2W 2 ∗

  • /8

= 0, µ + 3/(4W 2

∗ ) − 5Ω2

W 2

∗ + 2B2 ∗

  • /4

= 0.

−4 −2 2 4 0.1 0.2 0.3 0.4 0.5 x u1 and u2 (b) PDE ODE

Nonlinear Dynamical Systems – SDSU

LENCOS, Sevilla, Spain, 14-17 Jul 2009 – p.9/22

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Bifurcation of steady states: PDE vs ODE

1 2 3 4 5 6 −1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1 g B A C C B B D (a)

Unstable (ODE) Stable (ODE) Unstable (PDE) Stable (PDE)

Nonlinear Dynamical Systems – SDSU

LENCOS, Sevilla, Spain, 14-17 Jul 2009 – p.10/22

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Phase separation

Using pt A : gcr =

  • 6µ + 3
  • 15Ω2 + 4µ2
  • /
  • 26µ − 7
  • 15Ω2 + 4µ2
  • .

0.1 0.3 0.5 0.7 1 2 3 4 5 6 7 Ω g Stable separated state Stable mixed state Supercritical Pt. D (PDE) Subcritical Pt. A (ODE) Saddle−Node Pt. B (ODE)

Nonlinear Dynamical Systems – SDSU

LENCOS, Sevilla, Spain, 14-17 Jul 2009 – p.11/22

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Dynamics: oscillations trough each other

0.6 0.7 0.8 0.9 A (a) −1 1 B (b) 0.5 1 C+µt (c) −0.5 0.5 D (d) −0.2 0.2 E (e)

2 4 6 8 10 12 14 16 18 20 1 1.5 2 W t (f)

Nonlinear Dynamical Systems – SDSU

LENCOS, Sevilla, Spain, 14-17 Jul 2009 – p.12/22

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Dynamics: oscillations about fixed pts: separated osc.

0.5 0.6 0.7 A (a) −1 1 B (b) −2.5 −2 −1.5 −1 −0.5 C+µt (c) −0.5 0.5 D (d) −0.2 0.2 E (e)

2 4 6 8 10 12 14 16 18 20 0.8 1 1.2 1.4 W t (f)

Nonlinear Dynamical Systems – SDSU

LENCOS, Sevilla, Spain, 14-17 Jul 2009 – p.13/22

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Understanding the phase separation → Ueff

Newtonian reduction:

A(t) ≈ A∗ and W(t) ≈ W∗

Newton oscillations

d2B dt2 = −dUeff(B) dB ,

inside the effective potential

Ueff = Ω2 2 B2 + √ 2A2

∗g

2 e

− B2

2W 2 ∗ .

which becomes double well for large enough g (g > gcr)

−5 5 0.5 1 1.5 2 2.5 3 3.5 4 4.5 B Ueff g=0 g=2.6 g=6 g=20

Nonlinear Dynamical Systems – SDSU

LENCOS, Sevilla, Spain, 14-17 Jul 2009 – p.14/22

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Bifurcation of steady states

0.5 1 1.5 2 2.5 3 3.5 4 4.5 −1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1 ∆ h g A B C D E

Nonlinear Dynamical Systems – SDSU

LENCOS, Sevilla, Spain, 14-17 Jul 2009 – p.15/22

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Dynamics of unstable mixed state (for g > gcr)

(b) −5 5 0.2 0.4

x |u1,2(x)| 2

t= 0 t=40

(c) −5 5 0.2 0.4

x |u1,2(x)| 2

t= 0 t=145

(e) −5 5 0.2 0.4

x |u1,2(x)|2

t=0 t=10000

Nonlinear Dynamical Systems – SDSU

LENCOS, Sevilla, Spain, 14-17 Jul 2009 – p.16/22

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Dynamics of (unstable) higher order states

(b) −5 5 0.2 0.4

x |u1,2(x)| 2

t= 0 t=74

(d) −5 5 0.2 0.4

x |u1,2(x)|2

t=0 t=15000

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

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Bifurcation scenario for asymmetric states (µ1 = µ2)

A B C D E F G H I J K L M N O P Q R S T 0.6 1 1.4 1.8 −0.1 µ1 Var(u1)−Var(u2) A B C,K,R D E H G F J I L N M P Q S T

Nonlinear Dynamical Systems – SDSU

LENCOS, Sevilla, Spain, 14-17 Jul 2009 – p.18/22

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Outlook & to do’s (I)

Problem motivated by the experiments from David Hall in binary 87Rb. [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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SLIDE 23

Outlook & to do’s (I)

Problem motivated by the experiments from David Hall in binary 87Rb. [cf. our work in PRL 99, 190402 (2007)]. We are developing a 2D and 3D versions of this work to capture: the ring dartboard patterns.

1 10 20 30 40 50 60 70 80 90 100 110 120

Exp.

  • Num. A
  • Num. B

Exp.

  • Num. A
  • Num. B

|1 |2 Nonlinear Dynamical Systems – SDSU

LENCOS, Sevilla, Spain, 14-17 Jul 2009 – p.19/22

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Outlook & to do’s (II)

Ring

  • scillation

dynamics [movie]

20 40 1 2 r (µm)

  • ptical depth

20 40 r (µm)

Nonlinear Dynamical Systems – SDSU

LENCOS, Sevilla, Spain, 14-17 Jul 2009 – p.20/22

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Outlook & to do’s (III)

Remarkable experimental results from the LENS group: two species BECs with tunable interspecies interactions [PRL 100, 210402 (2008)]

→ need to develop a more general ansatz for asymmetric states.

Nonlinear Dynamical Systems – SDSU

LENCOS, Sevilla, Spain, 14-17 Jul 2009 – p.21/22

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SLIDE 26

Outlook & to do’s (III)

Remarkable experimental results from the LENS group: two species BECs with tunable interspecies interactions [PRL 100, 210402 (2008)]

→ need to develop a more general ansatz for asymmetric states.

FIN... GRACIAS !

Nonlinear Dynamical Systems – SDSU

LENCOS, Sevilla, Spain, 14-17 Jul 2009 – p.21/22

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SLIDE 27

NLDS: Nonlinear Dynamical Systems @ SDSU

http://nlds.sdsu.edu/ [Graduate Programs]

MS/PhD in Appl. Mathematics with concentration in Dynamical Systems. Fall 2009: MATH-537 : Advanced Ordinary Differential Equations MATH-538 : Dynamical Systems & Chaos I MATH-636 : Mathematical Modeling Spring 2010: MATH-531 : Advanced Partial Differential Equations MATH-639 : Nonlinear Waves MATH-638 : Dynamical Systems & Chaos II Fall 2010: MATH-635 : Pattern Formation MATH-693A : Advanced Numerical Analysis MATH-797 : Research Spring 2011: MATH-799A : Thesis – Project

Nonlinear Dynamical Systems – SDSU

LENCOS, Sevilla, Spain, 14-17 Jul 2009 – p.22/22