Complex dynamics in normal form Hamiltonian systems Hiromitsu Harada - - PowerPoint PPT Presentation

Quantum chaos: fundamentals and applications

Complex dynamics in normal form Hamiltonian systems

17th March. 2015

1Department of Physics, Tokyo Metropolitan University

1

2Laboratoire de Math´

ematiques et Physique Th´ eorique, Universit´ e Fran¸ cois Rabelais de Tours

Hiromitsu Harada,1 Akira Shudo,1 Amaury Mouchet,2 and J´ er´ emy Le Deunff3

3Max Planck Institut f¨

ur Physik komplexer Systeme

2

Motivation

Resonance-assisted tunneling in a normal form system

tunneling splitting vs１/hbar with a island chain without island chains

1/~

∆E

• J. Le Deunff, A. Mouchet, and P. Schlagheck, Phys. Rev. E 88, 042927 (2013).

R e q −π −π/2 π/2 π Re p π/2 π Im p 0.0 0.5 1.0

C C C1 C0 Γin Γout Γ′

in

Γ′

• ut

3

Motivation

∆E = |AT |2δE

• J. Le Deunff, A. Mouchet, and P. Schlagheck, Phys. Rev. E 88, 042927 (2013).

A semiclassical formula for tunneling splitting: Necessary to know the topology and the imaginary action of complex trajectories. Resonance-assisted tunneling in a normal form system

AT = e−σ/2~ 2 sin((Sin − Sout)/2l~)

4

Divide connected wells into two separated wells and focus on a single well case first.

＋ ＝

Simpler model

doublet system

∞

q p

Phase space with ✏ = ⌘ = −2.

Hamiltonian :

5

Exact analysis of a normal form system

H = p2 2 + q2 2 + ✏ ✓p2 2 + q2 2 ◆2 + ⌘p2q2.

where is an integration constant.

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˙ Q + ˙ P 4⌘(Q − P) = p PQ = ˙ Q − ˙ P 4 + 4✏(Q + P) + 4⌘(Q + P). 4(Q + P) + (2✏ + 2⌘)(Q + P)2 = 2⌘(Q − P)2 + C,

C

This yields New coordinate : P := p2, Q := q2, Hamilton’s equations :

˙ Q = (2 + 2✏(Q + P) + 4⌘Q) p PQ,

˙ P = (−2 − 2✏(P + Q) − 4⌘P) p PQ.

Exact analysis of a normal form system

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The form of solution :

Exact analysis of a normal form system

✓(t) = arcsin ✓↵ + sn2(t, k) sn2(t, k) − − ⌘ ✏A1/2 ◆ .

A := 1 + (✏ + ⌘)C

q = ± v u u t1 2 A1/2 ✏ + ⌘ sin ✓(t) + ✓ A −⌘(✏ + ⌘) ◆1/2 cos ✓(t) − 1 ✏ + ⌘ ! , p = ± v u u t1 2 A1/2 ✏ + ⌘ sin ✓(t) − ✓ A −⌘(✏ + ⌘) ◆1/2 cos ✓(t) − 1 ✏ + ⌘ ! .

sn(t, k) : Jacobi elliptic sn function

K and K’ are the periods of sn function.

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Singularity structure(Riemann sheet)

×,×: divergence point of

• : zero point of

: cut

Im T Re T

× ×

q(t)

iK’ 2iK’ Time plane of q(t)

• q(t)

2K 4K

• ×

×

T

6K 8K

× ×

• ×

×

q p

Phase space with ✏ = ⌘ = −2.

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Topology of trajectory (single island chain case)

: cut

Im T Re T

× ×

iK’ 2iK’ Time plane of q(t)

• 2K

4K

• T
• ×: divergence point of
• : zero point of

q(t) q(t)

q p

Phase space with ✏ = ⌘ = −2.

10

Topology of trajectory (single island chain case)

×: divergence point of

• : zero point of

: cut

Im T Re T

× ×

q(t)

Time plane of q(t)

• q(t)
• T

iK’ 2iK’ 2K 4K

11

q p

Phase space with ✏ = ⌘ = −2.

Topology of trajectory (single island chain case)

×,×: divergence point of

• : zero point of

: cut

Im T Re T

× ×

q(t)

Time plane of q(t)

• q(t)
• ×

×

T

iK’ 2iK’ 2K 4K

q p

12

Topology of trajectory (single island chain case)

Phase space with ✏ = ⌘ = −2.

Imaginary actions for these topologically distinct paths are different.

×,×: divergence point of

• : zero point of

: cut

Im T Re T

× ×

q(t)

Time plane of q(t)

• q(t)
• ×

×

T

iK’ 2iK’ 2K 4K

q p

13

Topology of trajectory (single island chain case)

Phase space with ✏ = ⌘ = −2.

×,×: divergence point of

• : zero point of

: cut

Im T Re T

× ×

q(t)

Time plane of q(t)

• q(t)
• ×

×

T

Which is cheaper? In this case, the green one is the cheaper.

iK’ 2iK’ 2K 4K

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Topology of trajectory(double island chain case)

Phase space with ✏ = −2, ⌘ = −2.7, = 0.9, ! = 1.8.

×

Time plane of q(t)

T Im T Re T

×

H = 1 2(q2 + p2) + ✏ 4(q2 + p2)2 + 8 (q2 + p2)3 + ⌘q2p2 + !q4p4

×: divergence point of

• : zero point of

q(t) q(t)

• ×
• : cut

15

Topology of trajectory(double island chain case)

Phase space with ✏ = −2, ⌘ = −2.7, = 0.9, ! = 1.8.

×

Time plane of q(t)

T Im T Re T

×

H = 1 2(q2 + p2) + ✏ 4(q2 + p2)2 + 8 (q2 + p2)3 + ⌘q2p2 + !q4p4

×: divergence point of

• : zero point of

q(t) q(t)

• ×
• : cut

16

Topology of trajectory(double island chain case)

Phase space with ✏ = −2, ⌘ = −2.7, = 0.9, ! = 1.8.

×

Time plane of q(t)

T Im T Re T

×

H = 1 2(q2 + p2) + ✏ 4(q2 + p2)2 + 8 (q2 + p2)3 + ⌘q2p2 + !q4p4

× × : divergence point of

• : zero point of

× × × × × ×

q(t) q(t)

• ×
• : cut

×

17

3 possible imaginary actions.

Time plane of q(t)

T Im T Re T

× × × × × × ×

Phase space with ✏ = −2, ⌘ = −2.7, = 0.9, ! = 1.8.

H = 1 2(q2 + p2) + ✏ 4(q2 + p2)2 + 8 (q2 + p2)3 + ⌘q2p2 + !q4p4

× × ×: divergence point of

• : zero point of

Topology of trajectory(double island chain case)

× × × × × ×

q(t) q(t)

• ×
• : cut

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If we glue two simple systems to form a doublet, the divergence points for a simple system may merge, and then a direct tunneling path must be created.

＋ ＝

Relation to the doublet case

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Looks different but the same topology, so the same imaginary action.

Topology of trajectory(double island chain case)

Phase space with ✏ = −2, ⌘ = −2.7, = 0.9, ! = 1.8.

×

Time plane of q(t)

T Im T Re T

× × × × × × ×

H = 1 2(q2 + p2) + ✏ 4(q2 + p2)2 + 8 (q2 + p2)3 + ⌘q2p2 + !q4p4

× × ×: divergence point of

• : zero point of

× × × × × ×

q(t) q(t)

• ×
• : cut

20

Conclusion

• We obtained the exact solution of a simple normal form

Hamiltonian system, which allows us to examine the Riemann sheet structure and singularities in the complex plane analytically.

• We numerically studied complex singularity structures in

more general cases, and explored how the paths with different imaginary actions appear.

• 1. paths with different topology
• rbit on a torus can go to either to nearest neighboring tori or to

infinity. (take either "local train" or "plane", no "express", "Shinkansen" … ) Two origins of the paths with different imaginary actions:

• 2. resolution of degenerated paths due to symmetry breaking.