Matter wave vortices: the quantum Spirograph Ricardo Carretero, and - - PowerPoint PPT Presentation
Matter wave vortices: the quantum Spirograph Ricardo Carretero, and Rafael Navarro, P .G. Kevrekidis, D. Frantzeskakis, P . Torres, and D.S. Hall Nonlinear Dynamical Systems Group http://nlds.sdsu.edu/ Computational Science Research Center
Nonlinear Dynamical Systems – SDSU
Ricardo Carretero, and Rafael Navarro, P .G. Kevrekidis, D. Frantzeskakis, P . Torres, and D.S. Hall Nonlinear Dynamical Systems Group http://nlds.sdsu.edu/ Computational Science Research Center Department of Mathematics and Statistics San Diego State University
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Introduction Physics of BECs External trapping → controlling the dimensionality Motivation: physical experiments
Nonlinear Dynamical Systems – SDSU
Introduction Physics of BECs External trapping → controlling the dimensionality Motivation: physical experiments Vortex dynamics in a magnetic trap Vortex precession in magnetic trap Vortex-vortex interactions Reduced (ODE) dynamics for vortex clusters Vortex dipoles: the Spirograph Bifurcation analysis Extensions to 3 and 4 vortex interactions
Nonlinear Dynamical Systems – SDSU
Introduction Physics of BECs External trapping → controlling the dimensionality Motivation: physical experiments Vortex dynamics in a magnetic trap Vortex precession in magnetic trap Vortex-vortex interactions Reduced (ODE) dynamics for vortex clusters Vortex dipoles: the Spirograph Bifurcation analysis Extensions to 3 and 4 vortex interactions Bifurcation in the actual experiment (prelim results).
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1925 Bose & Einstein predicted that a gas at very low temperature undergoes quantum “freezing”.
T ↓ ⇒ vel. ↓ ⇒ de Broglie: λ = h/p ⇒ λ ↑ ⇒ coherence ⇒ all atoms enter the SAME quantum state
BEC is to matter what laser is to light (coherent) 5th state of matter (gas + liquid + solid + plasma + BEC)
Nonlinear Dynamical Systems – SDSU
1925 Bose & Einstein predicted that a gas at very low temperature undergoes quantum “freezing”.
T ↓ ⇒ vel. ↓ ⇒ de Broglie: λ = h/p ⇒ λ ↑ ⇒ coherence ⇒ all atoms enter the SAME quantum state
BEC is to matter what laser is to light (coherent) 5th state of matter (gas + liquid + solid + plasma + BEC) 1995 Cornell + Wieman + others @ JILA + NIST + UC) achieved temperatures < 1/170M oK to produce a BEC (rubid- ium) for the 1st time.
movie (gif), movie (mov)
2001 Cornell + Ketterle + Wieman got the Nobel Prize in Physics for BECs. 2003 Abrikosov + Ginzburg + Leggett got the Nobel Prize in Physics for superconductors and superfluids. 2009 BEC count: Some 60 different BEC experiments.
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Close to T = 0 BEC → Gross-Pitaevskii Eq.:
i∂ψ ∂t =
2m∇2 + Vext(r) + gN |ψ|2
(1)
ψ(x, y, z, t): BEC wavefunction, |ψ|2: atom density, N: # of atoms, nonlinear coeff: g = 4π2as/m, as scattering length: as>0 : repulsive : (23Na, 87Rb, H, 4He, 85Rb)→ [DSs, vortices] as<0 : attractive : (7Li, 85Rb)→ [BSs, Bose Nova] Vext(r) = Vext(x, y, z) : external confining potential
Nonlinear Dynamical Systems – SDSU
Close to T = 0 BEC → Gross-Pitaevskii Eq.:
i∂ψ ∂t =
2m∇2 + Vext(r) + gN |ψ|2
(2)
ψ(x, y, z, t): BEC wavefunction, |ψ|2: atom density, N: # of atoms, nonlinear coeff: g = 4π2as/m, as scattering length: as>0 : repulsive : (23Na, 87Rb, H, 4He, 85Rb)→ [DSs, vortices] as<0 : attractive : (7Li, 85Rb)→ [BSs, Bose Nova] Vext(r) = Vext(x, y, z) : external confining potential
3D → 2D: pancake trap external (confining) magnetic potential:
Vext(x, y, z) = 1 2mω2
rr2 + 1
2mω2
zz2,
r2 = x2 + y2: in-plane dim., z: strong confining dir. ωz ≫ ωr : Quasi-2D BEC (pancake shaped) ⇒ vortices
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Take solution with topological charge S:
u(x, y, t) = f(r) eiSθ e−iµt
Nonlinear Dynamical Systems – SDSU
Take solution with topological charge S:
u(x, y, t) = f(r) eiSθ e−iµt
Vortex radial profile satisfies:
2r2
2rf ′ + 1 2f ′′ + |f|2f = 0
Nonlinear Dynamical Systems – SDSU
Take solution with topological charge S:
u(x, y, t) = f(r) eiSθ e−iµt
Vortex radial profile satisfies:
2r2
2rf ′ + 1 2f ′′ + |f|2f = 0
movie (gif),
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Precession frequency for a vortex inside a MT at distance
r from center [Fetter]:
(A ≈ 8.88).
ωpr = S ω0
pr
1 −
RT F
2
Nonlinear Dynamical Systems – SDSU
Precession frequency for a vortex inside a MT at distance
r from center [Fetter]:
(A ≈ 8.88).
ωpr = S ω0
pr
1 −
RT F
2 Precession frequency close to center ωpr ≈ S ω0
pr = Ω2
2µ ln
Ω
Nonlinear Dynamical Systems – SDSU
Precession frequency for a vortex inside a MT at distance
r from center [Fetter]:
(A ≈ 8.88).
ωpr = S ω0
pr
1 −
RT F
2 Precession frequency close to center ωpr ≈ S ω0
pr = Ω2
2µ ln
Ω
1000 −2 2 −1 1 2 t x c) y movie (gif),
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Opposite charge
⇒
[movie]
Nonlinear Dynamical Systems – SDSU
Opposite charge
⇒
[movie] Same charge
⇒
[movie]
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Movement induced by phase gradient and density gradient [Kivshar+Pismen+...]. One vortex induces gradients on the other vortex → movement
Nonlinear Dynamical Systems – SDSU
Movement induced by phase gradient and density gradient [Kivshar+Pismen+...]. One vortex induces gradients on the other vortex → movement Phase gradient of vortex phase eiθ is proportional to 1/separation2 and velocity is ⊥ to line joining other vortex (B = ωvort):
Nonlinear Dynamical Systems – SDSU
Movement induced by phase gradient and density gradient [Kivshar+Pismen+...]. One vortex induces gradients on the other vortex → movement Phase gradient of vortex phase eiθ is proportional to 1/separation2 and velocity is ⊥ to line joining other vortex (B = ωvort):
˙ x1 = − B S2 y1 − y2 2r2
12
, ˙ y1 = + B S2 x1 − x2 2r2
12
,
Nonlinear Dynamical Systems – SDSU
Movement induced by phase gradient and density gradient [Kivshar+Pismen+...]. One vortex induces gradients on the other vortex → movement Phase gradient of vortex phase eiθ is proportional to 1/separation2 and velocity is ⊥ to line joining other vortex (B = ωvort):
˙ x1 = − B S2 y1 − y2 2r2
12
, ˙ y1 = + B S2 x1 − x2 2r2
12
,
Superposition of N vortices:
˙ xm = −B
N
Sm ym − yn 2r2
mn
˙ ym = +B
N
Sm xm − xn 2r2
mn
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Let us add all contributions: vortex precession inside MT + vortex-vortex interactions
Nonlinear Dynamical Systems – SDSU
Let us add all contributions: vortex precession inside MT + vortex-vortex interactions Reduced ODE equations of motion for N vortices in MT (B = ωvort):
˙ xm = −Smωprym − B 2
N
Sn ym − yn r2
mn
˙ ym = Smωprxm + B 2
N
Sn xm − xn r2
mn
Nonlinear Dynamical Systems – SDSU
Let us add all contributions: vortex precession inside MT + vortex-vortex interactions Reduced ODE equations of motion for N vortices in MT (B = ωvort):
˙ xm = −Smωprym − B 2
N
Sn ym − yn r2
mn
˙ ym = Smωprxm + B 2
N
Sn xm − xn r2
mn
Conserved quantities: Hamiltonian and angular momentum:
H = −ω0
pr
2
N
ln(1 − r2
n) + B
4
N
N
SmSn ln(r2
mn),
L2 =
N
Snr2
n.
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240 300 360 420 60 120 180
a) b)
Coordinates in the trap (m)
445 ms 2 4 6 8 120 240 300 2 4 6 8 60
50 Pm
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60 120 180 240 300 360 420 445 ms
a) b) c)
50 Pm
Nonlinear Dynamical Systems – SDSU
60 120 180 240 300 360 420 445 ms
a) b) c)
50 Pm
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Equilibrium for diametrically opposed (symmetric) vortices:
req = 2
4 ω0
pr + B
Nonlinear Dynamical Systems – SDSU
Equilibrium for diametrically opposed (symmetric) vortices:
req = 2
4 ω0
pr + B
Linearize around equilibria: rotations with frequency:
ωeq = √ 2 ω0
pr
B 4 ω0
pr
3/2
−4 −2 2 4 −4 −2 2 4 xi yi 1 2 3 4 5 −4 −2 2 4 t x1, y1
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Consider diametrically opposed vortices but asymmetric wrt to the center: z1 = r1 exp(iωorbt) and z2 = −r2 exp(iωorbt) with r1 = r2. Asymmetric equilibrium distance:
ω0
pr
1 − r2
1
+ 1 1 − r2
2
B 2r1r2 = 0.
Nonlinear Dynamical Systems – SDSU
Consider diametrically opposed vortices but asymmetric wrt to the center: z1 = r1 exp(iωorbt) and z2 = −r2 exp(iωorbt) with r1 = r2. Asymmetric equilibrium distance:
ω0
pr
1 − r2
1
+ 1 1 − r2
2
B 2r1r2 = 0.
Rotating with freq:
ωorb = 1 2
pr(α − β) + γB
r1
r2 − r2 r1
where
α = 1 1 − r2
1
, β = 1 1 − r2
2
,
and
γ = 1 2r2
12
.
Nonlinear Dynamical Systems – SDSU
Consider diametrically opposed vortices but asymmetric wrt to the center: z1 = r1 exp(iωorbt) and z2 = −r2 exp(iωorbt) with r1 = r2. Asymmetric equilibrium distance:
ω0
pr
1 − r2
1
+ 1 1 − r2
2
B 2r1r2 = 0.
Rotating with freq:
ωorb = 1 2
pr(α − β) + γB
r1
r2 − r2 r1
where
α = 1 1 − r2
1
, β = 1 1 − r2
2
,
and
γ = 1 2r2
12
.
In co-rot frame: perturbs about equilibria result in rotating orbits.
→ epitrochoidal motion (spirograph) in the original frame !!! → Generic motion: quasi-periodic epitrochoids.
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Remember the experimental picture:
60 120 180 240 300 360 420 445 ms
a) b) c)
50 Pm
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On the original (lab.) frame:
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On the original (lab.) frame: On the co-rotating (ωorb) reference frame:
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Adimensionalize: X =
x RT F ,
τ = Ω2
2µ ln
A µ
Ω
c ≡ 1
2 ωvort ω0
pr .
Transform co-rot to polar: Xn =rn cos θn, Yn =rn sin θn, δmn = θm − θn :
˙ rm = −c rn sin δmn rmn , ˙ δmn =
m − r2 n
c cos δmn
rm rn r2
mn
+ 1 1 − r2
m
− 1 1 − r2
n
.
Nonlinear Dynamical Systems – SDSU
Adimensionalize: X =
x RT F ,
τ = Ω2
2µ ln
A µ
Ω
c ≡ 1
2 ωvort ω0
pr .
Transform co-rot to polar: Xn =rn cos θn, Yn =rn sin θn, δmn = θm − θn :
˙ rm = −c rn sin δmn rmn , ˙ δmn =
m − r2 n
c cos δmn
rm rn r2
mn
+ 1 1 − r2
m
− 1 1 − r2
n
.
Steady state is r1 = r2 = r∗ (for ANY r∗) and θ1 − θ2 = π, in co-rot:
ωorb ≡ ˙ θ1 = ˙ θ2 = c 2r2
∗
+ 1 1 − r2
∗
.
Nonlinear Dynamical Systems – SDSU
Adimensionalize: X =
x RT F ,
τ = Ω2
2µ ln
A µ
Ω
c ≡ 1
2 ωvort ω0
pr .
Transform co-rot to polar: Xn =rn cos θn, Yn =rn sin θn, δmn = θm − θn :
˙ rm = −c rn sin δmn rmn , ˙ δmn =
m − r2 n
c cos δmn
rm rn r2
mn
+ 1 1 − r2
m
− 1 1 − r2
n
.
Steady state is r1 = r2 = r∗ (for ANY r∗) and θ1 − θ2 = π, in co-rot:
ωorb ≡ ˙ θ1 = ˙ θ2 = c 2r2
∗
+ 1 1 − r2
∗
.
Perturbations about equilibria: rm = r∗ + Rm and δmn = π + ∆m Eqs on the pert.:
¨ Rm = −ω2
ep (Rn − Rm) ,
¨ ∆m = −ω2
ep (∆m − ∆n) ,
ω2
ep
= c2 2r4
∗
− 2c (1 − r2
∗)2 ,
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If r∗ < rcrit ≡
√c+ √ 2 ⇒ Epitrochoidal motion with freq ωep.
If r∗ > rcrit ⇒ INSTABILITY! Can it be observed in experiment?
Nonlinear Dynamical Systems – SDSU
If r∗ < rcrit ≡
√c+ √ 2 ⇒ Epitrochoidal motion with freq ωep.
If r∗ > rcrit ⇒ INSTABILITY! Can it be observed in experiment? Consider ASYMMETRIC equilibria: r1 = r2 and δ12 = θ1 − θ2 = π, where
ωasym
= r1 r∗
2 (r∗ 2 + r∗ 1)2 +
c (r∗
2 + r∗ 1)2 +
1 1 − r∗2
2
and radii r1 = r∗
1 and r2 = r∗ 2 satisfying
−r∗
1r∗ 2(r∗ 1 + r∗ 2)2 + c
1
1 − r∗2
2
Nonlinear Dynamical Systems – SDSU
If r∗ < rcrit ≡
√c+ √ 2 ⇒ Epitrochoidal motion with freq ωep.
If r∗ > rcrit ⇒ INSTABILITY! Can it be observed in experiment? Consider ASYMMETRIC equilibria: r1 = r2 and δ12 = θ1 − θ2 = π, where
ωasym
= r1 r∗
2 (r∗ 2 + r∗ 1)2 +
c (r∗
2 + r∗ 1)2 +
1 1 − r∗2
2
and radii r1 = r∗
1 and r2 = r∗ 2 satisfying
−r∗
1r∗ 2(r∗ 1 + r∗ 2)2 + c
1
1 − r∗2
2
These equilibria will have again epitrochoidal motion with freq:
ωep = 2c (r∗
1 + r∗ 2)
(r∗
1 + r∗ 2)2 −
1 2r∗2
1
− 1 2r∗2
2
+ r∗
1r∗ 2
c (1 − r∗2
1 )2 +
r∗
1r∗ 2
c (1 − r∗2
2 )2 .
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Angular momentum L2
0 = r2 1 + r2 2 and tan φ = r2/r1 (polar coord.)
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−0.5 0.5 −0.5 0.5 x y −0.5 0.5 −0.5 0.5 x y
200 400 −0.2 0.2 0.4
t L0
2 & H
−0.5 0.5 −0.5 0.5 x y −0.5 0.5 −0.5 0.5 x y
200 400 −0.2 0.2 0.4
t L0
2 & H
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−0.5 0.5 −0.5 0.5 x y −0.5 0.5 −0.5 0.5 x y
50 100 −0.2 0.2 0.4
t L0
2 & H
−0.5 0.5 −0.5 0.5 x y −0.5 0.5 −0.5 0.5 x y
200 400 −0.2 0.2 0.4
t L0
2 & H
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−0.5 0.5 −0.5 0.5 x y −0.5 0.5 −0.5 0.5 x y
200 400 −0.2 0.2 0.4
t L0
2 & H
−0.5 0.5 −0.5 0.5 x y −0.5 0.5 −0.5 0.5 x y
100 200 −0.2 0.2 0.4
t L0
2 & H
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−0.5 0.5 −0.5 0.5 x y −0.5 0.5 −0.5 0.5 x y
50 100 −0.2 0.2 0.4
t L0
2 & H
−0.5 0.5 −0.5 0.5 x y −0.5 0.5 −0.5 0.5 x y
200 400 −0.2 0.2 0.4
t L0
2 & H
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Reduced dynamics: experiment → PDE → ODE is very good! Match/understand dynamics in experiment Predict types of behavior: bifurcations
Nonlinear Dynamical Systems – SDSU
Reduced dynamics: experiment → PDE → ODE is very good! Match/understand dynamics in experiment Predict types of behavior: bifurcations Higher number of vortices (in progress...)
Nv = 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
Reduced dynamics: experiment → PDE → ODE is very good! Match/understand dynamics in experiment Predict types of behavior: bifurcations Higher number of vortices (in progress...)
Nv = 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
Reduced dynamics: experiment → PDE → ODE is very good! Match/understand dynamics in experiment Predict types of behavior: bifurcations Higher number of vortices (in progress...)
Nv = 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?
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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
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We have 6 variables − 2 conserved quantities − co-rotating frame
⇒ 4 degrees of freedom → possibility of chaos!
Nonlinear Dynamical Systems – SDSU
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
r1 = r2 = r3 = r∗ and θ1 − θ2 = θ2 − θ3 = θ3 − θ1 = 2π/3 with freq: ωorb = c r2
∗
+ 1 1 − r2
∗
Nonlinear Dynamical Systems – SDSU
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
r1 = r2 = r3 = r∗ and θ1 − θ2 = θ2 − θ3 = θ3 − θ1 = 2π/3 with freq: ωorb = c r2
∗
+ 1 1 − r2
∗
Epitrochoidal freq. about the equilibrium: (if r∗ < r(3)
crit)
ω2
ep = c2
r4
∗
− 2c (1 − r2
∗)2 .
Nonlinear Dynamical Systems – SDSU
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
r1 = r2 = r3 = r∗ and θ1 − θ2 = θ2 − θ3 = θ3 − θ1 = 2π/3 with freq: ωorb = c r2
∗
+ 1 1 − r2
∗
Epitrochoidal freq. about the equilibrium: (if r∗ < r(3)
crit)
ω2
ep = c2
r4
∗
− 2c (1 − r2
∗)2 .
Eigenvalues: (0, 0, 0, −ω2
ep, −ω2 ep, −ω2 ep) with eigenvectors:
= (0, 0, 0, 1, 1, 1),
= (1, 1, 1, 0, 0, 0),
= (0, 0, 0, −1, 0, 1),
= (0, 0, 0, −1, 1, 0),
= (−1, 0, 1, 0, 0, 0),
= (−1, 1, 0, 0, 0, 0).
LENCOS, Sevilla, July 2012. – p. 73/81
Nonlinear Dynamical Systems – SDSU
Radial perturbation Angular perturbation
−0.05 0.05 −0.05 0.05 x y −0.05 0.05 −0.05 0.05 x y
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Nonlinear Dynamical Systems – SDSU
0 = r2 1 + r2 2 + r2 3, tan θ =
1 + r2 2/r3 (spher. coord.)
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Nonlinear Dynamical Systems – SDSU
Use spherical coordinates: φ = arctan(r2/r1) and θ = arccos(r3/L0)
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Nonlinear Dynamical Systems – SDSU
L0 = 0.9 → L0 = 0.95 ⇒ ASYMMETRIC steady states!
LENCOS, Sevilla, July 2012. – p. 77/81
Nonlinear Dynamical Systems – SDSU
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Nonlinear Dynamical Systems – SDSU
Symmetric rotating solutions for r1 = r2 = r3 = r4 = r∗ and
θ1 − θ2 = θ2 − θ3 = θ3 − θ4 = θ4 − θ1 = π/2 with freq: ωorb = 3c 2r2
∗
+ 1 1 − r2
∗
.
Nonlinear Dynamical Systems – SDSU
Symmetric rotating solutions for r1 = r2 = r3 = r4 = r∗ and
θ1 − θ2 = θ2 − θ3 = θ3 − θ4 = θ4 − θ1 = π/2 with freq: ωorb = 3c 2r2
∗
+ 1 1 − r2
∗
.
TWO epitrochoidal freqs. about the equilibrium (if r∗ < r(4)
crit):
ω2
ep,a
= 3c (1 − r2
∗)2 − 9c2
4r4
∗
, ω2
ep,b
= 4c (1 − r2
∗)2 − 2c2
r4
∗
,
Nonlinear Dynamical Systems – SDSU
Symmetric rotating solutions for r1 = r2 = r3 = r4 = r∗ and
θ1 − θ2 = θ2 − θ3 = θ3 − θ4 = θ4 − θ1 = π/2 with freq: ωorb = 3c 2r2
∗
+ 1 1 − r2
∗
.
TWO epitrochoidal freqs. about the equilibrium (if r∗ < r(4)
crit):
ω2
ep,a
= 3c (1 − r2
∗)2 − 9c2
4r4
∗
, ω2
ep,b
= 4c (1 − r2
∗)2 − 2c2
r4
∗
,
Eigenvalues: (0, 0, −ω2
ep,a, −ω2 ep,a, −ω2 ep,b, −ω2 ep,b, −ω2 ep,b, −ω2 ep,b) with:
= (0, 0, 0, 0, 1, 1, 1, 1),
= (1, 1, 1, 1, 0, 0, 0, 0),
= (0, 0, 0, 0, −1, 1, −1, 1),
= (−1, 1, −1, 1, 0, 0, 0, 0, ),
= (0, 0, 0, 0, 0, −1, 0, 1),
= (0, 0, 0, 0, −1, 0, 1, 0),
= (0, −1, 0, 1, 0, 0, 0, 0),
= (−1, 0, 1, 0, 0, 0, 0, 0).
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Nonlinear Dynamical Systems – SDSU
−0.05 0.05 −0.05 0.05 x y −0.05 0.05 −0.05 0.05 x y −0.05 0.05 −0.05 0.05 x y −0.05 0.05 −0.05 0.05 x y
Radial pert.
ωep,a
Angular pert.
ωep,a
Radial pert.
ωep,b
Angular pert.
ωep,b
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Nonlinear Dynamical Systems – SDSU
Angular momentum L2
0 = r2 1 + r2 2 + r2 3 + r2 4 and tan θ = r1/r2
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