Vortex dynamics in parabolically trapped BECs Nonlinear Dynamical Systems – SDSU LENCOS, Sevilla, July 2012. – p. 47/81
BEC vortex: a single one without external potential Take solution with topological charge S : u ( x, y, t ) = f ( r ) e iSθ e − iµt Nonlinear Dynamical Systems – SDSU
BEC vortex: a single one without external potential Take solution with topological charge S : u ( x, y, t ) = f ( r ) e iSθ e − iµt Vortex radial profile satisfies: µ − S 2 � � f + 1 2 rf ′ + 1 2 f ′′ + | f | 2 f = 0 2 r 2 Nonlinear Dynamical Systems – SDSU
BEC vortex: a single one without external potential Take solution with topological charge S : u ( x, y, t ) = f ( r ) e iSθ e − iµt Vortex radial profile satisfies: µ − S 2 � � f + 1 2 rf ′ + 1 2 f ′′ + | f | 2 f = 0 2 r 2 movie (gif), Nonlinear Dynamical Systems – SDSU LENCOS, Sevilla, July 2012. – p. 48/81
BEC vortex: precession in the magnetic trap (MT) Precession frequency for a S ω 0 pr ω pr = vortex inside a MT at distance � 2 � r r from center [Fetter]: 1 − R T F ( A ≈ 8 . 88 ). Nonlinear Dynamical Systems – SDSU
BEC vortex: precession in the magnetic trap (MT) Precession frequency for a S ω 0 pr ω pr = vortex inside a MT at distance � 2 � r r from center [Fetter]: 1 − R T F ( A ≈ 8 . 88 ). pr = Ω 2 A µ � � Precession frequency close to center ω pr ≈ S ω 0 2 µ ln Ω Nonlinear Dynamical Systems – SDSU
BEC vortex: precession in the magnetic trap (MT) Precession frequency for a S ω 0 pr ω pr = vortex inside a MT at distance � 2 � r r from center [Fetter]: 1 − R T F ( A ≈ 8 . 88 ). pr = Ω 2 A µ � � Precession frequency close to center ω pr ≈ S ω 0 2 µ ln Ω c) 2 1 0 y −1 2 1000 0 x 500 t −2 0 movie (gif), Nonlinear Dynamical Systems – SDSU LENCOS, Sevilla, July 2012. – p. 49/81
BEC vortex interactions: pairwise dynamics Opposite charge [ movie ] ⇒ Nonlinear Dynamical Systems – SDSU
BEC vortex interactions: pairwise dynamics Opposite charge [ movie ] ⇒ Same charge ⇒ [ movie ] Nonlinear Dynamical Systems – SDSU LENCOS, Sevilla, July 2012. – p. 50/81
BEC vortex pairs: vortex-vortex interactions Movement induced by phase gradient and density gradient [Kivshar+Pismen+...]. One vortex induces gradients on the other vortex → movement Nonlinear Dynamical Systems – SDSU
BEC vortex pairs: vortex-vortex interactions Movement induced by phase gradient and density gradient [Kivshar+Pismen+...]. One vortex induces gradients on the other vortex → movement Phase gradient of vortex phase e iθ is proportional to 1 / separation 2 and velocity is ⊥ to line joining other vortex ( B = ω vort ) : Nonlinear Dynamical Systems – SDSU
BEC vortex pairs: vortex-vortex interactions Movement induced by phase gradient and density gradient [Kivshar+Pismen+...]. One vortex induces gradients on the other vortex → movement Phase gradient of vortex phase e iθ is proportional to 1 / separation 2 and velocity is ⊥ to line joining other vortex ( B = ω vort ) : y 1 − y 2 x 1 = − B S 2 ˙ , 2 r 2 12 x 1 − x 2 y 1 = + B S 2 ˙ , 2 r 2 12 Nonlinear Dynamical Systems – SDSU
BEC vortex pairs: vortex-vortex interactions Movement induced by phase gradient and density gradient [Kivshar+Pismen+...]. One vortex induces gradients on the other vortex → movement Phase gradient of vortex phase e iθ is proportional to 1 / separation 2 and velocity is ⊥ to line joining other vortex ( B = ω vort ) : y 1 − y 2 x 1 = − B S 2 ˙ , 2 r 2 12 x 1 − x 2 y 1 = + B S 2 ˙ , 2 r 2 12 Superposition of N vortices: N y m − y n � x m ˙ = − B S m 2 r 2 mn n =1 N x m − x n � y m ˙ = + B S m 2 r 2 mn n =1 Nonlinear Dynamical Systems – SDSU LENCOS, Sevilla, July 2012. – p. 51/81
BEC vortices in MT: reduced ODE dynamics Let us add all contributions: vortex precession inside MT + vortex -vortex interactions Nonlinear Dynamical Systems – SDSU
BEC vortices in MT: reduced ODE dynamics Let us add all contributions: vortex precession inside MT + vortex -vortex interactions Reduced ODE equations of motion for N vortices in MT ( B = ω vort ) : N − S m ω pr y m − B y m − y n � x m ˙ = S n r 2 2 mn n =1 N S m ω pr x m + B x m − x n � y m ˙ = S n r 2 2 mn n =1 Nonlinear Dynamical Systems – SDSU
BEC vortices in MT: reduced ODE dynamics Let us add all contributions: vortex precession inside MT + vortex -vortex interactions Reduced ODE equations of motion for N vortices in MT ( B = ω vort ) : N − S m ω pr y m − B y m − y n � x m ˙ = S n r 2 2 mn n =1 N S m ω pr x m + B x m − x n � y m ˙ = S n r 2 2 mn n =1 Conserved quantities: Hamiltonian and angular momentum: N N N − ω 0 n ) + B pr � � � ln(1 − r 2 S m S n ln( r 2 H = mn ) , 2 4 n =1 n =1 m � = n N � L 2 S n r 2 = n . 0 n =1 Nonlinear Dynamical Systems – SDSU LENCOS, Sevilla, July 2012. – p. 52/81
Vortex pairs inside MT: OPPOSITE charge pair: S 1 = 1 & S 2 = − 1 Nonlinear Dynamical Systems – SDSU LENCOS, Sevilla, July 2012. – p. 53/81
Experiments (David Hall) vs. theory a) 50 P m 0 60 120 180 240 b) 8 120 60 6 300 Coordinates in the trap ( � m) 4 2 360 0 0 0 2 420 4 6 240 300 8 445 ms Nonlinear Dynamical Systems – SDSU LENCOS, Sevilla, July 2012. – p. 54/81
More experiments (David Hall) → the Spirograph 50 P m a) b) c) 0 60 120 180 240 300 360 420 445 ms Nonlinear Dynamical Systems – SDSU
More experiments (David Hall) → the Spirograph 50 P m a) b) c) 0 60 120 180 240 300 360 420 445 ms Nonlinear Dynamical Systems – SDSU LENCOS, Sevilla, July 2012. – p. 55/81
Stationary (non-rotating) equilibria: Equilibrium for diametrically opposed (symmetric) vortices: � B r eq = 2 4 ω 0 pr + B Nonlinear Dynamical Systems – SDSU
Stationary (non-rotating) equilibria: Equilibrium for diametrically opposed (symmetric) vortices: � B r eq = 2 4 ω 0 pr + B Linearize around equilibria: rotations with frequency: � 3 / 2 √ � B 2 ω 0 ω eq = 1 + pr 4 ω 0 pr 4 4 2 2 x 1 , y 1 0 y i 0 −2 −2 −4 −4 0 1 2 3 4 5 −4 −2 0 2 4 t x i Nonlinear Dynamical Systems – SDSU LENCOS, Sevilla, July 2012. – p. 56/81
Asymmetric rotating equilibria: Consider diametrically opposed vortices but asymmetric wrt to the center: z 1 = r 1 exp( iω orb t ) and z 2 = − r 2 exp( iω orb t ) with r 1 � = r 2 . Asymmetric equilibrium distance: � � 1 1 B ω 0 + − = 0 . pr 1 − r 2 1 − r 2 2 r 1 r 2 1 2 Nonlinear Dynamical Systems – SDSU
Asymmetric rotating equilibria: Consider diametrically opposed vortices but asymmetric wrt to the center: z 1 = r 1 exp( iω orb t ) and z 2 = − r 2 exp( iω orb t ) with r 1 � = r 2 . Asymmetric equilibrium distance: � � 1 1 B ω 0 + − = 0 . pr 1 − r 2 1 − r 2 2 r 1 r 2 1 2 Rotating with freq: � � r 1 �� ω orb = 1 − r 2 ω 0 pr ( α − β ) + γB , 2 r 2 r 1 where 1 1 1 and α = , β = , γ = . 1 − r 2 1 − r 2 2 r 2 1 2 12 Nonlinear Dynamical Systems – SDSU
Asymmetric rotating equilibria: Consider diametrically opposed vortices but asymmetric wrt to the center: z 1 = r 1 exp( iω orb t ) and z 2 = − r 2 exp( iω orb t ) with r 1 � = r 2 . Asymmetric equilibrium distance: � � 1 1 B ω 0 + − = 0 . pr 1 − r 2 1 − r 2 2 r 1 r 2 1 2 Rotating with freq: � � r 1 �� ω orb = 1 − r 2 ω 0 pr ( α − β ) + γB , 2 r 2 r 1 where 1 1 1 and α = , β = , γ = . 1 − r 2 1 − r 2 2 r 2 1 2 12 In co -rot frame: perturbs about equilibria result in rotating orbits. → epitrochoidal motion (spirograph) in the original frame !!! → Generic motion: quasi-periodic epitrochoids. Nonlinear Dynamical Systems – SDSU LENCOS, Sevilla, July 2012. – p. 57/81
Motion about asymmetric equilibria: epitrochoids! Remember the experimental picture: 50 P m a) b) c) 0 60 120 180 240 300 360 420 445 ms Nonlinear Dynamical Systems – SDSU LENCOS, Sevilla, July 2012. – p. 58/81
Motion about asymmetric equilibria: epitrochoids! On the original (lab.) frame: Nonlinear Dynamical Systems – SDSU
Motion about asymmetric equilibria: epitrochoids! On the original (lab.) frame: On the co -rotating ( ω orb ) reference frame: Nonlinear Dynamical Systems – SDSU LENCOS, Sevilla, July 2012. – p. 59/81
Vortex pairs inside MT: SAME charge pair: S 1 = 1 = S 2 Nonlinear Dynamical Systems – SDSU LENCOS, Sevilla, July 2012. – p. 60/81
Adim. and transform to co-rot polar coord.: τ = Ω 2 A µ x c ≡ 1 ω vort 2 µ ln � � Adimensionalize: X = R T F , t, pr . ω 0 Ω 2 Transform co -rot to polar: X n = r n cos θ n , Y n = r n sin θ n , δ mn = θ m − θ n : − c r n sin δ mn r m ˙ = , r mn � c cos δ mn 1 1 ˙ r 2 m − r 2 � δ mn = + − . n r m r n r 2 1 − r 2 1 − r 2 mn m n Nonlinear Dynamical Systems – SDSU
Adim. and transform to co-rot polar coord.: τ = Ω 2 A µ x c ≡ 1 ω vort 2 µ ln � � Adimensionalize: X = R T F , t, pr . ω 0 Ω 2 Transform co -rot to polar: X n = r n cos θ n , Y n = r n sin θ n , δ mn = θ m − θ n : − c r n sin δ mn r m ˙ = , r mn � c cos δ mn 1 1 ˙ r 2 m − r 2 � δ mn = + − . n r m r n r 2 1 − r 2 1 − r 2 mn m n Steady state is r 1 = r 2 = r ∗ (for ANY r ∗ ) and θ 1 − θ 2 = π , in co-rot: c 1 ω orb ≡ ˙ θ 1 = ˙ θ 2 = + . 2 r 2 1 − r 2 ∗ ∗ Nonlinear Dynamical Systems – SDSU
Adim. and transform to co-rot polar coord.: τ = Ω 2 A µ x c ≡ 1 ω vort 2 µ ln � � Adimensionalize: X = R T F , t, pr . ω 0 Ω 2 Transform co -rot to polar: X n = r n cos θ n , Y n = r n sin θ n , δ mn = θ m − θ n : − c r n sin δ mn r m ˙ = , r mn � c cos δ mn 1 1 ˙ r 2 m − r 2 � δ mn = + − . n r m r n r 2 1 − r 2 1 − r 2 mn m n Steady state is r 1 = r 2 = r ∗ (for ANY r ∗ ) and θ 1 − θ 2 = π , in co-rot: c 1 ω orb ≡ ˙ θ 1 = ˙ θ 2 = + . 2 r 2 1 − r 2 ∗ ∗ Perturbations about equilibria: r m = r ∗ + R m and δ mn = π + ∆ m Eqs on the pert.: ¨ − ω 2 R m = ep ( R n − R m ) , ¨ − ω 2 ∆ m = ep (∆ m − ∆ n ) , c 2 2 c ω 2 = − ∗ ) 2 , ep 2 r 4 (1 − r 2 ∗ Nonlinear Dynamical Systems – SDSU LENCOS, Sevilla, July 2012. – p. 61/81
Epitrochoids about SYM. & ASYM. rotating equil.: √ c � If r ∗ < r crit ≡ 2 ⇒ Epitrochoidal motion with freq ω ep . √ √ c + If r ∗ > r crit ⇒ INSTABILITY! Can it be observed in experiment? Nonlinear Dynamical Systems – SDSU
Epitrochoids about SYM. & ASYM. rotating equil.: √ c � If r ∗ < r crit ≡ 2 ⇒ Epitrochoidal motion with freq ω ep . √ √ c + If r ∗ > r crit ⇒ INSTABILITY! Can it be observed in experiment? Consider ASYMMETRIC equilibria: r 1 � = r 2 and δ 12 = θ 1 − θ 2 = π , where r 1 c 1 ω asym = 1 ) 2 + 1 ) 2 + orb 1 − r ∗ 2 r ∗ 2 ( r ∗ 2 + r ∗ ( r ∗ 2 + r ∗ 2 and radii r 1 = r ∗ 1 and r 2 = r ∗ 2 satisfying 2 ) 2 + c 1 − r ∗ 2 1 − r ∗ 2 − r ∗ 1 r ∗ 2 ( r ∗ 1 + r ∗ � � � � = 0 . 1 2 Nonlinear Dynamical Systems – SDSU
Epitrochoids about SYM. & ASYM. rotating equil.: √ c � If r ∗ < r crit ≡ 2 ⇒ Epitrochoidal motion with freq ω ep . √ √ c + If r ∗ > r crit ⇒ INSTABILITY! Can it be observed in experiment? Consider ASYMMETRIC equilibria: r 1 � = r 2 and δ 12 = θ 1 − θ 2 = π , where r 1 c 1 ω asym = 1 ) 2 + 1 ) 2 + orb 1 − r ∗ 2 r ∗ 2 ( r ∗ 2 + r ∗ ( r ∗ 2 + r ∗ 2 and radii r 1 = r ∗ 1 and r 2 = r ∗ 2 satisfying 2 ) 2 + c 1 − r ∗ 2 1 − r ∗ 2 − r ∗ 1 r ∗ 2 ( r ∗ 1 + r ∗ � � � � = 0 . 1 2 These equilibria will have again epitrochoidal motion with freq: � r ∗ 1 r ∗ r ∗ 1 r ∗ 2 c 2 1 1 2 2 ω ep = 2 ) 2 − − + 1 ) 2 + 2 ) 2 . ( r ∗ 1 + r ∗ 2 r ∗ 2 2 r ∗ 2 2 ) ( r ∗ 1 + r ∗ c (1 − r ∗ 2 c (1 − r ∗ 2 1 2 Nonlinear Dynamical Systems – SDSU LENCOS, Sevilla, July 2012. – p. 62/81
Bifurcation of equilibria vs. ang. momentum: 2 vortice Angular momentum L 2 0 = r 2 1 + r 2 2 and tan φ = r 2 /r 1 (polar coord.) 0.75 a) c=0.1 0.5 φ / π 0.25 0 −0.25 0.4 0.6 0.8 1 1.2 1.4 L 0 Nonlinear Dynamical Systems – SDSU LENCOS, Sevilla, July 2012. – p. 63/81
EXPERIMENTAL results: hunting for the pitchfork bifurcation Nonlinear Dynamical Systems – SDSU LENCOS, Sevilla, July 2012. – p. 64/81
0.5 0.5 0.4 2 & H 0.2 0 0 x x L 0 0 −0.5 −0.5 −0.2 0.5 0 −0.5 0.5 0 −0.5 0 200 400 t y y 0.5 0.5 0.4 2 & H 0.2 0 0 x x L 0 0 −0.5 −0.5 −0.2 0 200 400 0.5 0 −0.5 0.5 0 −0.5 Nonlinear Dynamical Systems – SDSU t y y LENCOS, Sevilla, July 2012. – p. 65/81
0.5 0.5 0.4 2 & H 0.2 0 0 x x L 0 0 −0.5 −0.5 −0.2 0 50 100 0.5 0 −0.5 0.5 0 −0.5 t y y 0.5 0.5 0.4 2 & H 0.2 0 0 x x L 0 0 −0.5 −0.5 −0.2 0.5 0 −0.5 0.5 0 −0.5 0 200 400 t Nonlinear Dynamical Systems – SDSU y y LENCOS, Sevilla, July 2012. – p. 66/81
0.5 0.5 0.4 2 & H 0.2 0 0 x x L 0 0 −0.5 −0.5 −0.2 0.5 0 −0.5 0.5 0 −0.5 0 200 400 t y y 0.5 0.5 0.4 2 & H 0.2 0 0 x x L 0 0 −0.5 −0.5 −0.2 0.5 0 −0.5 0.5 0 −0.5 0 100 200 Nonlinear Dynamical Systems – SDSU t y y LENCOS, Sevilla, July 2012. – p. 67/81
0.5 0.5 0.4 2 & H 0.2 0 0 x x L 0 0 −0.5 −0.5 −0.2 0.5 0 −0.5 0.5 0 −0.5 0 50 100 t y y 0.5 0.5 0.4 2 & H 0.2 0 0 x x L 0 0 −0.5 −0.5 −0.2 0.5 0 −0.5 0.5 0 −0.5 0 200 400 Nonlinear Dynamical Systems – SDSU t y y LENCOS, Sevilla, July 2012. – p. 68/81
Recap / outlook Reduced dynamics: experiment → PDE → ODE is very good! Match/understand dynamics in experiment Predict types of behavior: bifurcations Nonlinear Dynamical Systems – SDSU
Recap / outlook Reduced dynamics: experiment → PDE → ODE is very good! Match/understand dynamics in experiment Predict types of behavior: bifurcations Higher number of vortices (in progress...) N v = 3 effective d.o.f. is 3 so possibility of chaos Epitrochoids for N -gons: multi-spirographs Celestial-type mechanics: periodic orbits Nonlinear Dynamical Systems – SDSU
Recap / outlook Reduced dynamics: experiment → PDE → ODE is very good! Match/understand dynamics in experiment Predict types of behavior: bifurcations Higher number of vortices (in progress...) N v = 3 effective d.o.f. is 3 so possibility of chaos Epitrochoids for N -gons: multi-spirographs Celestial-type mechanics: periodic orbits Many, many vortices (in progress...) Molecular dynamics Crystalization into vortex lattices Thermodynamics of vortex clusters → 2D quantum turbulence? Nonlinear Dynamical Systems – SDSU
Recap / outlook Reduced dynamics: experiment → PDE → ODE is very good! Match/understand dynamics in experiment Predict types of behavior: bifurcations Higher number of vortices (in progress...) N v = 3 effective d.o.f. is 3 so possibility of chaos Epitrochoids for N -gons: multi-spirographs Celestial-type mechanics: periodic orbits Many, many vortices (in progress...) Molecular dynamics Crystalization into vortex lattices Thermodynamics of vortex clusters → 2D quantum turbulence? Higher dimensions (in progress...) Vortex rings interactions 3D quantum turbulence? Nonlinear Dynamical Systems – SDSU LENCOS, Sevilla, July 2012. – p. 69/81
END... GRACIAS! Nonlinear Dynamical Systems – SDSU LENCOS, Sevilla, July 2012. – p. 70/81
NLDS: Nonlinear Dynamical Systems @ SDSU http://nlds.sdsu.edu/ [Graduate Programs] MS in Appl. Mathematics with concentration in Dynamical Systems. Fall Year 1: MATH -537 : Advanced Ordinary Differential Equations MATH-538 : Dynamical Systems & Chaos I MATH-636 : Mathematical Modeling Spring Year 1: MATH-531 : Advanced Partial Differential Equations MATH-639 : Nonlinear Waves MATH-638 : Dynamical Systems & Chaos II Fall Year 2: MATH-635 : Pattern Formation MATH-693A : Advanced Numerical Analysis MATH-797 : Research Spring Year 2: MATH-799A : Thesis – Project Nonlinear Dynamical Systems – SDSU LENCOS, Sevilla, July 2012. – p. 71/81
THREE vortices with equal charge S 1 = S 2 = S 3 = +1 Nonlinear Dynamical Systems – SDSU LENCOS, Sevilla, July 2012. – p. 72/81
THREE vortices: We have 6 variables − 2 conserved quantities − co -rotating frame ⇒ 4 degrees of freedom → possibility of chaos! Nonlinear Dynamical Systems – SDSU
THREE vortices: We have 6 variables − 2 conserved quantities − co -rotating frame ⇒ 4 degrees of freedom → possibility of chaos! We can still compute symmetric rotating solutions for r 1 = r 2 = r 3 = r ∗ and θ 1 − θ 2 = θ 2 − θ 3 = θ 3 − θ 1 = 2 π/ 3 with freq: ω orb = c 1 + r 2 1 − r 2 ∗ ∗ Nonlinear Dynamical Systems – SDSU
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