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.