Univerzita Hradec Královéchaos17.uhk.cz/Slides/shudo.pdf · Completely integrable Nonintegrable Stable and unstable manifolds in the real phase space Amphibious Complex orbits and

Completely integrable Nonintegrable

Stable and unstable manifolds in the real phase space

Amphibious Complex orbits and Dynamical Tunneling両生複素軌道と動的トンネル効果

Akira Shudo (TMU)

Collaboration with K.S. Ikeda (Ritsumeikan), Y. Ishii (Kyushu), A. Ishikawa (TMU),Y. Hanada and A. Tanaka (TMU)

stable manifold unstable manifold


Y. Hanada (2014)

Stable and unstable manifolds in the real phase space

Amphibious Complex orbits and Dynamical Tunneling

Akira Shudo (TMU)

in honor of Professor Petr Seba’s 60th Birthday

Collaboration with K.S. Ikeda (Ritsumeikan), Y. Ishii (Kyushu), A. Ishikawa (TMU),Y. Hanada and A. Tanaka (TMU)

Amphibious Complex orbits and Dynamical Tunneling

Akira Shudo (TMU)

in honor of Professor Petr Seba’s 60th Birthday

1D systems

H =p2

2+ V(q)

Classically, always completely integrable.

2D systems

H =p2

1

2+

p22

2+ V(q1, q2)

Classically, nonintegrable in general.

Confinement occurs not only due to energy barriers, but also dynamical barriersrestrict the classical motion

configuration space-2

-1

0

1

2

-1.0- 0.5

0.00.5

1.0

0.0

0.5

1.0

1.5

2.02

� 2 � 1 1 2

� 1.0

� 0.5

0.5

1.0

1D systems

H =p2

2+ V(q)

Classically, always complete integrable.

2D systems

H =p2

1

2+

p22

2+ V(q1, q2)


Close analogy between Dynamical localization and Anderson localization

Still exponentially small. Why flooding does not occur ?

Quantum tunneling in 1D and 2D systems

Hamiltonian systems in 1D and 2D

両生複素軌道と動的トンネル効果

Poincare section of phase space

1D systems

H =p2

2+ V(q)

Classically, always completely integrable.

2D systems

H =p2

1

2+

p22

2+ V(q1, q2)


Confinement occurs not only due to energy barriers, but also dynamical barriersrestrict the classical motion

configuration space-2

-1

0

1

2

-1.0- 0.5

0.00.5

1.0

0.0

0.5

1.0

1.5

2.02

� 2 � 1 1 2

� 1.0

� 0.5

0.5

1.0

1D systems

H =p2

2+ V(q)


2D systems

H =p2

1

2+

p22

2+ V(q1, q2)








x

-V(x)

x

-V(x)

Discrete maps (more generally)

Forbidden process in classical dynamics

Aa!

F−n(Bb) = ∅ for ∀n, ifAa,Bb are dynamically separated.

F :"

p′q′

#=

"p − V′(q)

q + p′

#

Semiclassical approach

Complex paths and quantum tunneling Instanton

Quantum tunneling in double well potentialInstanton

it

x

Quantum tunneling in multidimensions

1D double well potential


Quantum tunneling and complex paths

Flooding induced by destructing coherence in the chaotic region

Why flooding does not occur ?

Amphibious nature of complex orbits

- Exponentially many complex orbits connect regular states a and chaoticstates b.

- The orbits behave as regular orbits when wander stay in the regular region,while they become chaotic after reaching the chaotic sea.

-

Complex orbits contributing to semiclassical propagator

γ : classical orbits connecting a and b

γ is approximated by Ws(p)

γ′ is approximated by Ws(p′)

γ′′ is approximated by Ws(p′′)







-






x

-V(x)

x

-V(x)

Discrete maps (more generally)


Aa!

F−n(Bb) = ∅ for ∀n, ifAa,Bb are dynamically separated.

F :"

p′q′

#=

"p − V′(q)

q + p′

#

Semiclassical approach

Complex paths and quantum tunneling Instanton

Quantum tunneling in double well potentialInstanton

it

x










-












-















-













-






1D systems

H =p2

2+ V(q)


2D systems

H =p2

1

2+

p22

2+ V(q1, q2)

Classically, nonintegrable in general.configuration space









- Confinement occurs not only due to energy barriers, but also dynamical barriers

- Chaos appears in phase space












-













-






1D systems

H =p2

2+ V(q)


2D systems

H =p2

1

2+

p22

2+ V(q1, q2)

Classically, nonintegrable in general.configuration space









- Confinement occurs not only due to energy barriers, but also dynamical barriers

- Chaos appears in phase space



Classical dynamics in mixed phase space

Area-preserving map

F :!

p′q′

"=

!p − V′(q)

q + p′

"


Aa ∩ F−n(Bb) = ∅ for ∀n, ifAa,Bb(∈ R) are dynamically separated.

Quantum map

K(a, b) = ⟨b|Un|a⟩ =#· · ·

# $

j

dqj

$

j

dpj exp%

i!

S({qj}, {pj})&

Quantum map

U = e−i!T(p)e−

i!V(q)

Propagator

K(a, b) = ⟨b|Un|a⟩ =#· · ·

# $

j

dqj

$

j

dpj exp%

i!

S({qj}, {pj})&

Tunneling process in quantum dynamics

K(a, b) ! 0 even ifAa,Bb(∈ R) are dynamically separated.q p

s ! 0，β = ∞ s ! 0，β = 2 s ! 0，β = 0.1


Area-preserving map

F :!

p′q′

"=

!p − V′(q)

q + p′

"



Aa Bb


Aarea-preserving map

F :!

p′q′

"=

!p − V′(q)

q + p′

"



Quantum map

K(a, b) = ⟨b|Un|a⟩ =#· · ·# $

j

dqj

$

j

dpj exp%

i!

S({qj}, {pj})&

Quantum map

U = e−i!T(p)e−

i!V(q)

Propagator

K(a, b) = ⟨b|Un|a⟩ =#· · ·# $

j

dqj

$

j

dpj exp%

i!

S({qj}, {pj})&





F :!

p′q′

"=

!p − V′(q)

q + p′

"



Quantum map

K(a, b) = ⟨b|Un|a⟩ =#· · ·# $

j

dqj

$

j

dpj exp%

i!

S({qj}, {pj})&

Quantum map

U = e−i!T(p)e−

i!V(q)

Propagator

K(a, b) = ⟨b|Un|a⟩ =#· · ·# $

j

dqj

$

j

dpj exp%

i!

S({qj}, {pj})&



Aa Bb

Quantum Tunneling in Nonintegrable systems

Akira Shudo (TMU)

Area-preserving map and mixed phase space


Area-preserving map

F :!

p′q′

"=

!p − V′(q)

q + p′

"



Quantum map

K(a, b) = ⟨b|Un|a⟩ =#· · ·

# $

j

dqj

$

j

dpj exp%

i!

S({qj}, {pj})&

Quantum map

U = e−i!T(p)e−

i!V(q)

Propagator

K(a, b) = ⟨b|Un|a⟩ =#· · ·

# $

j

dqj

$

j

dpj exp%

i!

S({qj}, {pj})&



s ! 0，β = ∞ s ! 0，β = 2 s ! 0，β = 0.1


Area-preserving map

F :!

p′q′

"=

!p − V′(q)

q + p′

"



Aa Bb



F :!

p′q′

"=

!p − V′(q)

q + p′

"



Quantum map

K(a, b) = ⟨b|Un|a⟩ =#· · ·# $

j

dqj

$

j

dpj exp%

i!

S({qj}, {pj})&

Quantum map

U = e−i!T(p)e−

i!V(q)

Propagator

K(a, b) = ⟨b|Un|a⟩ =#· · ·# $

j

dqj

$

j

dpj exp%

i!

S({qj}, {pj})&





F :!

p′q′

"=

!p − V′(q)

q + p′

"



Quantum map

K(a, b) = ⟨b|Un|a⟩ =#· · ·# $

j

dqj

$

j

dpj exp%

i!

S({qj}, {pj})&

Quantum map

U = e−i!T(p)e−

i!V(q)

Propagator

K(a, b) = ⟨b|Un|a⟩ =#· · ·# $

j

dqj

$

j

dpj exp%

i!

S({qj}, {pj})&



Aa Bb


Akira Shudo (TMU)




F :!

p′q′

"=

!p − V′(q)

q + p′

"



Quantum map

K(a, b) = ⟨b|Un|a⟩ =#· · ·# $

j

dqj

$

j

dpj exp%

i!

S({qj}, {pj})&

Quantum map

U = e−i!T(p)e−

i!V(q)

Propagator

K(a, b) = ⟨b|Un|a⟩ =#· · ·# $

j

dqj

$

j

dpj exp%

i!

S({qj}, {pj})&


K(a, b) ! 0 even ifAa,Bb(∈ R) are dynamically separated.


Area-preserving map

F :!

p′q′

"=

!p − V′(q)

q + p′

"



Aa Bb



F :!

p′q′

"=

!p − V′(q)

q + p′

"



Quantum map

K(a, b) = ⟨b|Un|a⟩ =#· · ·# $

j

dqj

$

j

dpj exp%

i!

S({qj}, {pj})&

Quantum map

U = e−i!T(p)e−

i!V(q)

Propagator

K(a, b) = ⟨b|Un|a⟩ =#· · ·# $

j

dqj

$

j

dpj exp%

i!

S({qj}, {pj})&





F :!

p′q′

"=

!p − V′(q)

q + p′

"



Quantum map

K(a, b) = ⟨b|Un|a⟩ =#· · ·# $

j

dqj

$

j

dpj exp%

i!

S({qj}, {pj})&

Quantum map

U = e−i!T(p)e−

i!V(q)

Propagator

K(a, b) = ⟨b|Un|a⟩ =#· · ·# $

j

dqj

$

j

dpj exp%

i!

S({qj}, {pj})&



Aa Bb


Akira Shudo (TMU)



Area-preserving map

F :!

p′q′

"=

!p − V′(q)

q + p′

"



Aa Bb



F :!

p′q′

"=

!p − V′(q)

q + p′

"



Quantum map

K(a, b) = ⟨b|Un|a⟩ =#· · ·# $

j

dqj

$

j

dpj exp%

i!

S({qj}, {pj})&

Quantum map

U = e−i!T(p)e−

i!V(q)

Propagator

K(a, b) = ⟨b|Un|a⟩ =#· · ·# $

j

dqj

$

j

dpj exp%

i!

S({qj}, {pj})&





F :!

p′q′

"=

!p − V′(q)

q + p′

"



Quantum map

K(a, b) = ⟨b|Un|a⟩ =#· · ·# $

j

dqj

$

j

dpj exp%

i!

S({qj}, {pj})&

Quantum map

U = e−i!T(p)e−

i!V(q)

Propagator

K(a, b) = ⟨b|Un|a⟩ =#· · ·# $

j

dqj

$

j

dpj exp%

i!

S({qj}, {pj})&



Aa Bb


Akira Shudo (TMU)


Area-presering map

F :!

p′q′

"=

!p + V′(q)q + T′(p′)

"

Quantum propagator

K(a, b) = ⟨b|Un|a⟩ =# +∞

−∞· · ·# +∞

−∞

$

j

dqj

$

j

dpj exp%

i!

S({qj}, {pj})&

|a⟩ : initial state |b⟩ : final state

Semiclassical approximation (Van-Vleck, Gutzwiller)

Ksc(a, b) ='

γ

A(γ)n (a, b) exp

( i!

S(γ)n (a, b)

)

A = { (p, q) ∈ C2 | A(p, q) = a } : initial set

B = { (p, q) ∈ C2 | B(p, q) = b } : final set

If Fn(A) ∩ B = ∅, then γ should be complex.

Area-presering map

F :!

p′q′

"=

!p + V′(q)q + T′(p′)

"

Quantum propagator

K(a, b) = ⟨b|Un|a⟩ =# +∞

−∞· · ·# +∞

−∞

$

j

dqj

$

j

dpj exp%

i!

S({qj}, {pj})&



Ksc(a, b) ='

γ

A(γ)n (a, b) exp

( i!

S(γ)n (a, b)

)




γ: classical orbits connecting a and b

Area-presering map

F :!

p′q′

"=

!p + V′(q)q + T′(p′)

"

Quantum propagator

K(a, b) = ⟨b|Un|a⟩ =# +∞

−∞· · ·# +∞

−∞

$

j

dqj

$

j

dpj exp%

i!

S({qj}, {pj})&



Ksc(a, b) ='

γ

A(γ)n (a, b) exp

( i!

S(γ)n (a, b)

)




2-D autonomous Hamiltonian H(q1, q2, p1, p2)

Constant of motion: H(q1, q2, p1, p2) = E

Poincare map F : Σ !→ Σ is area-preserving (symplectic)

Action of the closed loops C

S[C] =!

Cp · dq − Hdt

Difference of action of arbitrary closed loops C and C′

S[C] − S[C′] ="

T∇ × A · d2s (A = (p, 0,−H))

Since ∇ × A = −(q, p,−1), and the Hamiltonian vector field is parallel to the tube T ,

S[C] = S[C′]

On the constant energy surface

S[C] =!

Cp · dq

instanton





S[C] =!

Cp · dq − Hdt


S[C] − S[C′] ="

T∇ × A · d2s (A = (p, 0,−H))


S[C] = S[C′]


S[C] =!

Cp · dq

Completely integrable Nonintegrableinstanton

Y. Hanada (2014)

(Greene, Percival, Berretti, Marmi, · · · )





S[C] =!

Cp · dq − Hdt


S[C] − S[C′] ="

T∇ × A · d2s (A = (p, 0,−H))


S[C] = S[C′]


S[C] =!

Cp · dq


Complexified invariant manifold and natural boundaries

instanton

Y. Hanada (2014)


Natural boundary

As β (resp. M) increases, invariant curves obtained by integrable approxima-tion well approximate KAM curves on R2.

Turning points on R2 or close to R2 become dense, which makes the leadingsemiclassical approximation invalid.

With an increase in β, invariant curves for the integrable map (g = 0, s = 0)well approximate KAM curves on R2.

Analytic continuation ofan invariant curve





S[C] =!

Cp · dq − Hdt


S[C] − S[C′] ="

T∇ × A · d2s (A = (p, 0,−H))


S[C] = S[C′]


S[C] =!

Cp · dq

instanton





S[C] =!

Cp · dq − Hdt


S[C] − S[C′] ="

T∇ × A · d2s (A = (p, 0,−H))


S[C] = S[C′]


S[C] =!

Cp · dq

instanton

Y. Hanada (2014)






S[C] =!

Cp · dq − Hdt


S[C] − S[C′] ="

T∇ × A · d2s (A = (p, 0,−H))


S[C] = S[C′]


S[C] =!

Cp · dq


Y. Hanada (2014)






S[C] =!

Cp · dq − Hdt


S[C] − S[C′] ="

T∇ × A · d2s (A = (p, 0,−H))


S[C] = S[C′]


S[C] =!

Cp · dq


Y. Hanada (2014)






S[C] =!

Cp · dq − Hdt


S[C] − S[C′] ="

T∇ × A · d2s (A = (p, 0,−H))


S[C] = S[C′]


S[C] =!

Cp · dq


Complexified invariant manifold and natural boundaries

instanton

Y. Hanada (2014)


How are disconnected regions connected ?

Complex dynamics in 1-dimensional maps

1-dimensional maps F : C !→ C

F : z′ = F(z)

Classify the orbits according to the behavior of n → ∞

I = { z ∈ C | limn→∞

Fn(z) = ∞ } : Set of escaping points

K = { z ∈ C | limn→∞

Fn(z) is bounded } : Filled Julia set

In particular

J = ∂K : Julia set

F = C − J : Fatou set

Note: Alternative definitions for FP, JP based on

(1) normal family, (2) equicontinuity, (3) density of periodic oribts



F : z′ = f (z)


IP = { z ∈ C | limn→∞

Pn(z) = ∞ } : Set of escaping points

KP = { z ∈ C | limn→∞

Pn(z) is bounded } : Filled Julia set

In particular

JP = ∂KP : Julia set

FP = C − JP : Fatou set





F : z′ = F(z)


I = { z ∈ C | limn→∞


K = { z ∈ C | limn→∞


In particular







F : z′ = f (z)


IP = { z ∈ C | limn→∞


KP = { z ∈ C | limn→∞


In particular







F : z′ = F(z)


I = { z ∈ C | limn→∞


K = { z ∈ C | limn→∞


In particular







F : z′ = f (z)


IP = { z ∈ C | limn→∞


KP = { z ∈ C | limn→∞


In particular





Julia set in 1-D dynamics

・ F(z) = z2

I = { |z| > 1 }, K = { |z| ≤ 1 }, J = { |z| = 1 }

- z = 0 and z = ∞ are both attracting fixed points of F.The points z ∈ I tend to∞ and also the points z ∈ K − J converge toz = 0 monotonically.

- The orbits z ∈ J are chaotic.Putting z = e2πiθ, then the map on J can be reduced to θ %→ 2θ (mod 1).


・ F(z) = z2

I = { |z| > 1 }, K = { |z| ≤ 1 }, J = { |z| = 1 }




・ F(z) = z2

I = { |z| > 1 }, K = { |z| ≤ 1 }, J = { |z| = 1 }



・ F(z) = z2 + c


・ F(z) = z2

I = { |z| > 1 }, K = { |z| ≤ 1 }, J = { |z| = 1 }

- z = 0 and z = ∞ are both attracting fixed points of F.The points z ∈ I tend to∞ and also the points z ∈ K − Jconverge to z = 0 monotonically.

- The orbits z ∈ J are chaotic.Putting z = e2πiθ, then the map on J can be reduced toθ %→ 2θ (mod 1).

・ F(z) = z2 + c

The behavior around z = 0

Theorem (Koenigs) F(z) is holomorphic near z = 0 and has theTaylor expansion

F(z) = λz + c2z2 + · · · (0 < |λ| < 1)

Then there exists a conformal map φ : U → C which satisfies thefunctional equation (Schroder equation)

φ!F(z)"= λφ(z) (z ∈ U)

where U is a neighborhood of z = 0.

Note : If |λ| > 1, then one can show the same assertion by consideringthe inverse function.

z = 0 z = ∞



F(z) = λz + c2z2 + · · · (0 < |λ| < 1)


φ!F(z)"= λφ(z) (z ∈ U)



z = 0 z = ∞


・ F(z) = z2

I = { |z| > 1 }, K = { |z| ≤ 1 }, J = { |z| = 1 }




・ F(z) = z2

I = { |z| > 1 }, K = { |z| ≤ 1 }, J = { |z| = 1 }




・ F(z) = z2

I = { |z| > 1 }, K = { |z| ≤ 1 }, J = { |z| = 1 }



・ F(z) = z2 + c


・ F(z) = z2

I = { |z| > 1 }, K = { |z| ≤ 1 }, J = { |z| = 1 }

- z = 0 and z = ∞ are both attracting fixed points of F.The points z ∈ I tend to∞ and also the points z ∈ K − Jconverge to z = 0 monotonically.

- The orbits z ∈ J are chaotic.Putting z = e2πiθ, then the map on J can be reduced toθ %→ 2θ (mod 1).

・ F(z) = z2 + c



F(z) = λz + c2z2 + · · · (0 < |λ| < 1)


φ!F(z)"= λφ(z) (z ∈ U)



z = 0 z = ∞



F(z) = λz + c2z2 + · · · (0 < |λ| < 1)


φ!F(z)"= λφ(z) (z ∈ U)



z = 0 z = ∞


Quantum tunneling in multidimension ?


2-dimensional maps F : C2 !→ C2

F :!

z′1

z′2

"=

!f (z1, z2)g(z1, z2)

"

Orbits are classified according to the behavior of n → ±∞

I± = { (z1, z2) ∈ C2 | limn→∞

F±n(z1, z2) = ∞ }K± = { (z1, z2) ∈ C2 | lim

n→∞F±n(z1, z2) is bounded in C2 }

In particular

K = K+ ∩ K− : filled Julia set

J± = ∂K± : forward (resp. backward) Julia set

J = J+ ∩ J− : Julia set








F :!

z′1

z′2

"=

!f (z1, z2)g(z1, z2)

"

Here the orbits are classified according to the behavior of n → ∞

I± = { (z1, z2) ∈ C2 | limn→∞

P±n(z1, z2) = ∞ }K± = { (z1, z2) ∈ C2 | lim

n→∞P±n(z1, z2) is bounded in C2 }

In particular

J = ∂K+ ∩ ∂K− : Julia set








F :!

z′1

z′2

"=

!f (z1, z2)g(z1, z2)

"


I± = { (z1, z2) ∈ C2 | limn→∞

F±n(z1, z2) = ∞ }K± = { (z1, z2) ∈ C2 | lim


In particular











F :!

z′1

z′2

"=

!f (z1, z2)g(z1, z2)

"


I± = { (z1, z2) ∈ C2 | limn→∞

P±n(z1, z2) = ∞ }K± = { (z1, z2) ∈ C2 | lim


In particular









F :!

z′1

z′2

"=

!f (z1, z2)g(z1, z2)

"


I± = { (z1, z2) ∈ C2 | limn→∞

F±n(z1, z2) = ∞ }K± = { (z1, z2) ∈ C2 | lim


In particular











F :!

z′1

z′2

"=

!f (z1, z2)g(z1, z2)

"


I± = { (z1, z2) ∈ C2 | limn→∞

P±n(z1, z2) = ∞ }K± = { (z1, z2) ∈ C2 | lim


In particular





Stable and unstable manifold theorem

F :!

p′q′

"=

!p + V′(q)

q + p′

"( V(q) : polynomial )

Theorem (Bedford-Smillie 1991) For any unstable periodic orbits p,

Ws(p) = J+ and Wu(p) = J−

where Ws(p) (resp.Wu(p)) denotes stable (resp. unstable) manifold for p and

J± = ∂K± is called the forward (backward) Julia set.

Here, K± = { (p, q) ∈ C2 | ∥Fn(p, q)∥ is bounded (n → ±∞) }

Note : Ws(p) and Wu(p) are both locally 1-dimensional complex (2-dimensional

real) manifold in C2.


F :!

p′q′

"=

!p + V′(q)

q + p′










F :!

p′q′

"=

!p + V′(q)

q + p′










F :!

p′q′

"=

!p + V′(q)

q + p′









Bedford E and Smillie J :Invent. Math. 103 (1991) 69-99; J. Amer. Math. Soc. 4 (1991) 657-679; Math. Ann. 294 (1992) 395-420;J.Geom.Anal. 8 (1998) 349-383; Annal. sci. de l′Ecole norm. super. 32 (1999) 455-497; American Journal o fMathematics 124 (2002) 221-271; Ann.Math. 148 (1998) 695-735; Ann.Math. 160 (2004) 1-26

Bedford E, Lyubich E and Smillie JInvent.Math. 112 (1993) 77-125; Invent.Math. 114 (1993) 277-288

Bedford E and Smillie J : J. Amer.Math. Soc. 4 (1991) 657-679Bedford E and Smillie J : Math. Ann. 294 (1992) 395-420Bedford E, Lyubich E and Smillie J : Invent.Math. 112 (1993) 77-125Bedford E, Lyubich E and Smillie J : Invent.Math. 114 (1993) 277-288Bedford E and Smillie J : J.Geom.Anal. 8 (1998) 349-383Bedford E and Smillie J : Annal. sci. de l′Ecole norm. super. 32 (1999) 455-497Bedford E and Smillie J : American Journal o f Mathematics 124 (2002) 221-271

Bedford E and Smillie J : Ann.Math. 148 (1998) 695-735Bedford E and Smillie J : Ann.Math. 160 (2004) 1-26


F :!

p′q′

"=

!p + V′(q)

q + p′










F :!

p′q′

"=

!p + V′(q)

q + p′









Bedford E and Smillie J :Invent. Math. 103 (1991) 69-99; J. Amer. Math. Soc. 4 (1991) 657-679; Math. Ann. 294 (1992) 395-420;J.Geom.Anal. 8 (1998) 349-383; Annal. sci. de l′Ecole norm. super. 32 (1999) 455-497; American Journal o fMathematics 124 (2002) 221-271; Ann.Math. 148 (1998) 695-735; Ann.Math. 160 (2004) 1-26

Bedford E, Lyubich E and Smillie JInvent.Math. 112 (1993) 77-125; Invent.Math. 114 (1993) 277-288

Bedford E and Smillie J : J. Amer.Math. Soc. 4 (1991) 657-679Bedford E and Smillie J : Math. Ann. 294 (1992) 395-420Bedford E, Lyubich E and Smillie J : Invent.Math. 112 (1993) 77-125Bedford E, Lyubich E and Smillie J : Invent.Math. 114 (1993) 277-288Bedford E and Smillie J : J.Geom.Anal. 8 (1998) 349-383Bedford E and Smillie J : Annal. sci. de l′Ecole norm. super. 32 (1999) 455-497Bedford E and Smillie J : American Journal o f Mathematics 124 (2002) 221-271

Bedford E and Smillie J : Ann.Math. 148 (1998) 695-735Bedford E and Smillie J : Ann.Math. 160 (2004) 1-26

p







F :!

p′q′

"=

!p + V′(q)

q + p′









p






Note :

1. This theorem holds even in the system with mixed phase space.

2. Ws(p) and Wu(p) are both locally 1-dimensional complex( = 2-dimensional real) manifold in C2.

p







F :!

p′q′

"=

!p + V′(q)

q + p′









Quantum tunneling in double wells







Amphibious complex orbits and flooding


- The orbits behave as regular orbits when they stay in the regular region,while they become chaotic after reaching the chaotic sea.

- Exponentially many complex orbits connect regular chaotic states and floodthe entire region.

-



-


initial setA

p p′ p′′

Ws(p) Ws(p′) Ws(p′′)

Since Ws(p) = J+ for any p,

Ws(p) =Ws(p′) =Ws(p′′) = · · ·

Dynamics (classical, quantum and semiclassical)Area-presering map

F :!

p′q′

"=

!p + V′(q)q + T′(p′)

"

Quantum propagator

K(a, b) = ⟨b|Un|a⟩ =# +∞

−∞· · ·# +∞

−∞

$

j

dqj

$

j

dpj exp%

i!

S({qj}, {pj})&



Ksc(a, b) ='

γ

A(γ)n (a, b) exp

( i!

S(γ)n (a, b)

)




initial setA natural boundary

p p′ p′′



Ws(p) =Ws(p′) =Ws(p′′) = · · ·


F :!

p′q′

"=

!p + V′(q)q + T′(p′)

"

Quantum propagator

K(a, b) = ⟨b|Un|a⟩ =# +∞

−∞· · ·# +∞

−∞

$

j

dqj

$

j

dpj exp%

i!

S({qj}, {pj})&



Ksc(a, b) ='

γ

A(γ)n (a, b) exp

( i!

S(γ)n (a, b)

)





p p′ p′′



Ws(p) =Ws(p′) =Ws(p′′) = · · ·


F :!

p′q′

"=

!p + V′(q)q + T′(p′)

"

Quantum propagator

K(a, b) = ⟨b|Un|a⟩ =# +∞

−∞· · ·# +∞

−∞

$

j

dqj

$

j

dpj exp%

i!

S({qj}, {pj})&



Ksc(a, b) ='

γ

A(γ)n (a, b) exp

( i!

S(γ)n (a, b)

)





p p′ p′′



Ws(p) =Ws(p′) =Ws(p′′) = · · ·


F :!

p′q′

"=

!p + V′(q)q + T′(p′)

"

Quantum propagator

K(a, b) = ⟨b|Un|a⟩ =# +∞

−∞· · ·# +∞

−∞

$

j

dqj

$

j

dpj exp%

i!

S({qj}, {pj})&



Ksc(a, b) ='

γ

A(γ)n (a, b) exp

( i!

S(γ)n (a, b)

)





p p′ p′′



Ws(p) =Ws(p′) =Ws(p′′) = · · ·

γ1 γ2 γ3

γ′1 γ′2 γ′3

γ′′1 γ′′2 γ′′3


F :!

p′q′

"=

!p + V′(q)q + T′(p′)

"

Quantum propagator

K(a, b) = ⟨b|Un|a⟩ =# +∞

−∞· · ·# +∞

−∞

$

j

dqj

$

j

dpj exp%

i!

S({qj}, {pj})&



Ksc(a, b) ='

γ

A(γ)n (a, b) exp

( i!

S(γ)n (a, b)

)





p p′ p′′



Ws(p) =Ws(p′) =Ws(p′′) = · · ·

γ1 γ2 γ3

γ′1 γ′2 γ′3

γ′′1 γ′′2 γ′′3


F :!

p′q′

"=

!p + V′(q)q + T′(p′)

"

Quantum propagator

K(a, b) = ⟨b|Un|a⟩ =# +∞

−∞· · ·# +∞

−∞

$

j

dqj

$

j

dpj exp%

i!

S({qj}, {pj})&



Ksc(a, b) ='

γ

A(γ)n (a, b) exp

( i!

S(γ)n (a, b)

)





p p′ p′′



Ws(p) =Ws(p′) =Ws(p′′) = · · ·

γ1 γ2 γ3

γ′1 γ′2 γ′3

γ′′1 γ′′2 γ′′3


F :!

p′q′

"=

!p + V′(q)q + T′(p′)

"

Quantum propagator

K(a, b) = ⟨b|Un|a⟩ =# +∞

−∞· · ·# +∞

−∞

$

j

dqj

$

j

dpj exp%

i!

S({qj}, {pj})&



Ksc(a, b) ='

γ

A(γ)n (a, b) exp

( i!

S(γ)n (a, b)

)





p p′ p′′



Ws(p) =Ws(p′) =Ws(p′′) = · · ·

γ1 γ2 γ3

γ′1 γ′2 γ′3

γ′′1 γ′′2 γ′′3


F :!

p′q′

"=

!p + V′(q)q + T′(p′)

"

Quantum propagator

K(a, b) = ⟨b|Un|a⟩ =# +∞

−∞· · ·# +∞

−∞

$

j

dqj

$

j

dpj exp%

i!

S({qj}, {pj})&



Ksc(a, b) ='

γ

A(γ)n (a, b) exp

( i!

S(γ)n (a, b)

)





p p′ p′′



Ws(p) =Ws(p′) =Ws(p′′) = · · ·

γ1 γ2 γ3

γ′1 γ′2 γ′3

γ′′1 γ′′2 γ′′3


F :!

p′q′

"=

!p + V′(q)q + T′(p′)

"

Quantum propagator

K(a, b) = ⟨b|Un|a⟩ =# +∞

−∞· · ·# +∞

−∞

$

j

dqj

$

j

dpj exp%

i!

S({qj}, {pj})&



Ksc(a, b) ='

γ

A(γ)n (a, b) exp

( i!

S(γ)n (a, b)

)





p p′ p′′



Ws(p) =Ws(p′) =Ws(p′′) = · · ·

γ1 γ2 γ3

γ′1 γ′2 γ′3

γ′′1 γ′′2 γ′′3


F :!

p′q′

"=

!p + V′(q)q + T′(p′)

"

Quantum propagator

K(a, b) = ⟨b|Un|a⟩ =# +∞

−∞· · ·# +∞

−∞

$

j

dqj

$

j

dpj exp%

i!

S({qj}, {pj})&



Ksc(a, b) ='

γ

A(γ)n (a, b) exp

( i!

S(γ)n (a, b)

)





p p′ p′′



Ws(p) =Ws(p′) =Ws(p′′) = · · ·

γ1 γ2 γ3

γ′1 γ′2 γ′3

γ′′1 γ′′2 γ′′3


F :!

p′q′

"=

!p + V′(q)q + T′(p′)

"

Quantum propagator

K(a, b) = ⟨b|Un|a⟩ =# +∞

−∞· · ·# +∞

−∞

$

j

dqj

$

j

dpj exp%

i!

S({qj}, {pj})&



Ksc(a, b) ='

γ

A(γ)n (a, b) exp

( i!

S(γ)n (a, b)

)





p p′ p′′



Ws(p) =Ws(p′) =Ws(p′′) = · · ·

γ1 γ2 γ3

γ′1 γ′2 γ′3

γ′′1 γ′′2 γ′′3


F :!

p′q′

"=

!p + V′(q)q + T′(p′)

"

Quantum propagator

K(a, b) = ⟨b|Un|a⟩ =# +∞

−∞· · ·# +∞

−∞

$

j

dqj

$

j

dpj exp%

i!

S({qj}, {pj})&



Ksc(a, b) ='

γ

A(γ)n (a, b) exp

( i!

S(γ)n (a, b)

)





p p′ p′′



Ws(p) =Ws(p′) =Ws(p′′) = · · ·

γ1 γ2 γ3

γ′1 γ′2 γ′3

γ′′1 γ′′2 γ′′3


F :!

p′q′

"=

!p + V′(q)q + T′(p′)

"

Quantum propagator

K(a, b) = ⟨b|Un|a⟩ =# +∞

−∞· · ·# +∞

−∞

$

j

dqj

$

j

dpj exp%

i!

S({qj}, {pj})&



Ksc(a, b) ='

γ

A(γ)n (a, b) exp

( i!

S(γ)n (a, b)

)




Note:the final form does not change if the smoothing function ‘ tanh’ is replaced by ‘ arctan’.Quasi-periodic kicked rotors

(Casati-Guarneri-Shepelyansky 1989, Chabe et al 2008)

(Hufnagel-Ketzmerick-Otto-Schanz, 2002, Backer-Ketzmerick-Monasta, 2005)

Arbitrary itineracy p→ p′ → p′′ → · · ·is possible after reaching R2.

Propagator:

K(a, b) = ⟨b|U|a⟩ =!· · ·

! "

j

dqj

"

j

exp#

i!

S({qj}, {pj})$

Semiclassical approximation

Ksc(a, b) =%

γ

A(γ)n (a, b) exp

#i!

S(γ)n (a, b)

$

Natural boundary

initial state a

Quantum tunneling in double wells







Amphibious complex orbits and flooding




-






Tunneling orbits and Julia sets

Semiclassical sum

Ksc(a, b) =!

γ

A(γ)n (a, b) exp

" i!

S(γ)n (a, b)

#

Theorem (AS, Y. Ishii and K.S. Ikeda) For polynomial maps F,

(i) If F is hyperbolic and htop(F|R2) = log 2, then C = J+

(ii) If F is hyperbolic and htop(F|R2) > 0, then C = J+

(iii) If htop(F|R2) > 0, then J+ ⊂ C ⊂ K+

Here htop(P|R2) is topological entropy confined on R2, and semiclassicallycontributing complex orbits are introduced as

C ≡ $(q, p) ∈M∞ | Im Sn(q, p) converges absolutely at (q, p)

%

( Proof ) apply the convergent theory of current (Bedford-Smillie)

Tunneling orbits and Julia sets

Semiclassical sum

Ksc(a, b) =!

γ

A(γ)n (a, b) exp

" i!

S(γ)n (a, b)

#

Theorem (AS, Y. Ishii and K.S. Ikeda) For polynomial maps F,

(i) If F is hyperbolic and htop(F|R2) = log 2, then C = J+

(ii) If F is hyperbolic and htop(F|R2) > 0, then C = J+

(iii) If htop(F|R2) > 0, then J+ ⊂ C ⊂ K+

Here htop(P|R2) is topological entropy confined on R2, and semiclassically con-tributing complex orbits are introduced as

C ≡ $(q, p) ∈M∞ | Im Sn(q, p) converges absolutely at (q, p)

%

( Proof ) apply the convergent theory of current (Bedford-Smillie)

Semiclassical wavefunction hypothesis

For generic non-integrable systems (mixed systems), eigenfunctions localizeexclusively on the ergodic or the integrable component.

Localized states (either on torus or chaotic region)

Amphibious states (extended over torus and chaotic region)

Semiclassical wavefunction hypothesis

For generic non-integrable systems (mixed systems), eigenfunctions localizeexclusively on the ergodic or the integrable component.

Localized states (either on torus or chaotic region)

Amphibious states (extended over torus and chaotic region)

dynamical localization Anderson localization

(momentum space) (con f iguration space)

delocalized localized critical

ε ξ n log Ptorusn ⟨p2⟩

“Ray and wave chaos in asymmetric resonant optical cavities”JU Nockel and AD Stone, Nature 385, 45 (1997)

N(p) p |ψ(p)|2

t = iτ

“Eigenstates ignores regular and chaotic phase space structures”L.Hufnagel, R.Ketzmerick, M.F.Otto and H.Schanz, PRL, 89, 154101-1 (2002)

“Flooding of chaotic eigenstates into regular phase space islands”A.Backer, R.Ketzmerick, A.G.Monasta, PRL, 94, 045102-1 (2005)

ε = 0.5

Snapshots of wavefunction— below and above the transition point —

Violation of semiclassical eigenfunction hypothesis ?






-






Area-preserving map

f :

⇤p�

q�

⌅=

⇤p � V �(q)q + T �(p�)

⌅

where

T �(p) = ap + d1 � d2

+1

2

�ap � � + d1

⇥tanh b(p � pd)

+1

2

��ap + � + d2

⇥tanh b(p + pd)

V �(q) = K sin q

f :

⇤p�

q�

⌅=

⇤p � V �(q)q + T �(p�)

⌅

where

T �(p) =1

2ap{tanh b(p + L) � tanh b(p � L)}

+1

2(ap � �){tanh b(p � pd) � tanh b(p + pd)}

V �(q) = K sin q

T �(p) = T �0(p) + ⇥T �

noise(p)

Area-preserving map

f :

⇤p�

q�

⌅=

⇤p � V �(q)q + T �(p�)

⌅

where

T �(p) = ap + d1 � d2

+1

2

�ap � � + d1

⇥tanh b(p � pd)

+1

2

��ap + � + d2

⇥tanh b(p + pd)

V �(q) = K sin q

f :

⇤p�

q�

⌅=

⇤p � V �(q)q + T �(p�)

⌅

where

T �(p) =1


+1


V �(q) = K sin q

T �(p) = T �0(p) + ⇥T �

noise(p)

-1

-0.8

-0.6

-0.4

-0.2

0

1.0×1075×1060

log 10

Pnto

rus

n

L=2π (cl)

initial state

Survival probability in the torus region:

P torusn =

� pb

pa

|�n(p)|2dp

pa pb


P torusn =

� pb

pa

|�n(p)|2dp

pa pb

Strong correlation between dynamical tunneling and dynamical localization

Survival probability in the regular region (classical) (quantum)

Ptorusn =

! pb

pa

Pn(p)dp

Ptorusn =

! pb

pa

|ψn(p)|2dp



Ptorusn =

! pb

pa

Pn(p)dp

Ptorusn =

! pb

pa

|ψn(p)|2dp



P torusn =

! pb

pa

Pn(p)dp

P torusn =

! pb

pa

|ψn(p)|2dp



P torusn =

! pb

pa

Pn(p)dp

P torusn =

! pb

pa

|ψn(p)|2dp






-






-1

-0.8

-0.6

-0.4

-0.2

0

1.0×1075×1060

log 10

Pnto

rus

n

L=2π (cl)

-1

-0.8

-0.6

-0.4

-0.2

0

1.0×1075×1060

log 10

Pnto

rus

n

L=2π (cl)L=∞ (qm)Area-preserving map

f :

⇤p�

q�

⌅=

⇤p � V �(q)q + T �(p�)

⌅

where

T �(p) = ap + d1 � d2

+1

2

�ap � � + d1

⇥tanh b(p � pd)

+1

2

��ap + � + d2

⇥tanh b(p + pd)

V �(q) = K sin q

f :

⇤p�

q�

⌅=

⇤p � V �(q)q + T �(p�)

⌅

where

T �(p) =1


+1


V �(q) = K sin q

T �(p) = T �0(p) + ⇥T �

noise(p)

Area-preserving map

f :

⇤p�

q�

⌅=

⇤p � V �(q)q + T �(p�)

⌅

where

T �(p) = ap + d1 � d2

+1

2

�ap � � + d1

⇥tanh b(p � pd)

+1

2

��ap + � + d2

⇥tanh b(p + pd)

V �(q) = K sin q

f :

⇤p�

q�

⌅=

⇤p � V �(q)q + T �(p�)

⌅

where

T �(p) =1


+1


V �(q) = K sin q

T �(p) = T �0(p) + ⇥T �

noise(p)

initial state


P torusn =

� pb

pa

|�n(p)|2dp

pa pb


P torusn =

� pb

pa

|�n(p)|2dp

pa pb



Ptorusn =

! pb

pa

Pn(p)dp

Ptorusn =

! pb

pa

|ψn(p)|2dp



Ptorusn =

! pb

pa

Pn(p)dp

Ptorusn =

! pb

pa

|ψn(p)|2dp



P torusn =

! pb

pa

Pn(p)dp

P torusn =

! pb

pa

|ψn(p)|2dp



P torusn =

! pb

pa

Pn(p)dp

P torusn =

! pb

pa

|ψn(p)|2dp






-






-1

-0.8

-0.6

-0.4

-0.2

0

1.0×1075×1060

log 10

Pnto

rus

n

L=2π (cl)

-1

-0.8

-0.6

-0.4

-0.2

0

1.0×1075×1060

log 10

Pnto

rus

n

L=2π (cl)L=∞ (qm)

-1

-0.8

-0.6

-0.4

-0.2

0

1.0×1075×1060

log 10

Pnto

rus

n

L=2π (cl)L=∞ (qm)L=6π (qm)L=4π (qm)L=2π (qm)

Area-preserving map

f :

⇤p�

q�

⌅=

⇤p � V �(q)q + T �(p�)

⌅

where

T �(p) = ap + d1 � d2

+1

2

�ap � � + d1

⇥tanh b(p � pd)

+1

2

��ap + � + d2

⇥tanh b(p + pd)

V �(q) = K sin q

f :

⇤p�

q�

⌅=

⇤p � V �(q)q + T �(p�)

⌅

where

T �(p) =1


+1


V �(q) = K sin q

T �(p) = T �0(p) + ⇥T �

noise(p)

Area-preserving map

f :

⇤p�

q�

⌅=

⇤p � V �(q)q + T �(p�)

⌅

where

T �(p) = ap + d1 � d2

+1

2

�ap � � + d1

⇥tanh b(p � pd)

+1

2

��ap + � + d2

⇥tanh b(p + pd)

V �(q) = K sin q

f :

⇤p�

q�

⌅=

⇤p � V �(q)q + T �(p�)

⌅

where

T �(p) =1


+1


V �(q) = K sin q

T �(p) = T �0(p) + ⇥T �

noise(p)

L

initial state

noise

noise


P torusn =

� pb

pa

|�n(p)|2dp

pa pb


P torusn =

� pb

pa

|�n(p)|2dp

pa pb



Ptorusn =

! pb

pa

Pn(p)dp

Ptorusn =

! pb

pa

|ψn(p)|2dp



Ptorusn =

! pb

pa

Pn(p)dp

Ptorusn =

! pb

pa

|ψn(p)|2dp



P torusn =

! pb

pa

Pn(p)dp

P torusn =

! pb

pa

|ψn(p)|2dp



P torusn =

! pb

pa

Pn(p)dp

P torusn =

! pb

pa

|ψn(p)|2dp

SummarySummary

- Instanton disappears due to the presence of natural boundaries in genericmultidimensional systems.

- Classically disconnected regions are connected via orbits in the Julia set.

- The orbits in the Julia set are ergodic and have amphibious nature.

- Exponentially many orbits potentially exist behind the tunneling process fromregular to chaotic regions but they are blocked by dynamical localizationin the chaotic region.

- Flooding occurs if quantum interference effects are suppressed eitherby adding noise or by coupling the systems to invoke the Anderson transitionin the chaotic region.

Documents

Univerzita Hradec Královéchaos17.uhk.cz/Slides/shudo.pdf · Completely integrable Nonintegrable Stable and unstable manifolds in the real phase space Amphibious Complex orbits and