Quantum chaos: fundamentals and applications

Complex dynamics in normal form Hamiltonian systems

17th March. 2015

1Department of Physics, Tokyo Metropolitan University

1

2Laboratoire de Mathematiques et Physique Theorique, Universite Francois Rabelais de Tours

Hiromitsu Harada,1 Akira Shudo,1 Amaury Mouchet,2 and Jeremy Le Deun↵3

3Max Planck Institut fur Physik komplexer Systeme

2

Motivation

Resonance-assisted tunneling in a normal form system

tunneling splitting vs１/hbarwith a island chain

without island chains

1/~

�E

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

Re q

−π−π/2

0π/2

π

Re p

0

π/2

π

Imp

0.0

0.5

1.0

CC

C1

C0

ΓinΓoutΓ′

inΓ′

out

3

Motivation

�E = |AT |2�E

J. Le Deun↵, 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��/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

1

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.

6

Q+ P

4⌘(Q� P )=

pPQ =

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

7

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 = ±

vuut1

2

A1/2

✏ + ⌘sin ✓(t) +

✓A

�⌘(✏ + ⌘)

◆1/2

cos ✓(t)� 1

✏ + ⌘

!,

p = ±

vuut1

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.

8

Singularity structure(Riemann sheet)

×,×: divergence point of●: zero point of

: cutIm T

Re T

××

q(t)

iK’

2iK’

0

Time plane of q(t)

●

q(t)

2K 4K●

× ×

T

6K 8K× ×●

●

× ×

q

p

Phase space with ✏ = ⌘ = �2.

9

Topology of trajectory (single island chain case)

: cutIm T

Re T

××iK’

2iK’

0

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

: cutIm T

Re T

××

q(t)

0

Time plane of q(t)

●

q(t)

●

T

iK’

2iK’

0 2K 4K

11

q

p

Phase space with ✏ = ⌘ = �2.

Topology of trajectory (single island chain case)

×,×: divergence point of●: zero point of

: cutIm T

Re T

××

q(t)

0

Time plane of q(t)

●

q(t)

●

× ×

T

iK’

2iK’

0 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

: cutIm T

Re T

××

q(t)

0

Time plane of q(t)

●

q(t)

●

× ×

T

iK’

2iK’

0 2K 4K

q

p

13

Topology of trajectory (single island chain case)

Phase space with ✏ = ⌘ = �2.

×,×: divergence point of●: zero point of

: cutIm T

Re T

××

q(t)

0

Time plane of q(t)

●

q(t)

●

× ×

T

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

iK’

2iK’

0 2K 4K

14

Topology of trajectory(double island chain case)

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

×Time plane of q(t)

TIm 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)

TIm 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)

TIm 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)

TIm 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

18

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

19

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)

TIm 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

 orbit 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.