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

1 Quantum chaos: fundamentals and applications 17th March. 2015 Complex dynamics in normal form Hamiltonian systems Hiromitsu Harada , 1 Akira Shudo , 1 Amaury Mouchet , 2 and J´ emy Le Deun ff 3 er´ 1 Department of Physics , Tokyo Metropolitan University 2 Laboratoire de Math´ ematiques et Physique Th´ eorique, Universit´ e Fran¸ cois Rabelais de Tours 3 Max Planck Institut f¨ ur Physik komplexer Systeme

2 Motivation Resonance-assisted tunneling in a normal form system tunneling splitting vs １ /hbar with a island chain ∆ E without island chains 1 / ~ J. Le Deun ff , A. Mouchet, and P. Schlagheck, Phys. Rev. E 88 , 042927 (2013).

3 Motivation Resonance-assisted tunneling in a normal form system A semiclassical formula for C 0 tunneling splitting: 1 . 0 C Im p ∆ E = | A T | 2 δ E C 0 . 5 C 1 e − σ / 2 ~ 0 . 0 Γ in Γ out Γ ′ π A T = Γ ′ out in π / 2 2 sin(( S in − S out ) / 2 l ~ ) Re p − π − π / 2 0 R π / 2 e q 0 π J. Le Deun ff , A. Mouchet, and P. Schlagheck, Phys. Rev. E 88 , 042927 (2013). Necessary to know the topology and the imaginary action of complex trajectories.

4 ＋ Simpler model Divide connected wells into two separated wells and focus on a single well case first. doublet system ＝ ∞

5 Exact analysis of a normal form system Hamiltonian : p ◆ 2 H = p 2 2 + q 2 ✓ p 2 2 + q 2 + ⌘ p 2 q 2 . 2 + ✏ 2 q Phase space with ✏ = ⌘ = − 2.

6 Exact analysis of a normal form system New coordinate : P := p 2 , Q := q 2 , ˙ p Hamilton’s equations : Q = (2 + 2 ✏ ( Q + P ) + 4 ⌘ Q ) PQ, ˙ p P = ( − 2 − 2 ✏ ( P + Q ) − 4 ⌘ P ) PQ. Q + ˙ ˙ Q − ˙ ˙ P P p 4 ⌘ ( Q − P ) = PQ = 4 + 4 ✏ ( Q + P ) + 4 ⌘ ( Q + P ) . This yields 4( Q + P ) + (2 ✏ + 2 ⌘ )( Q + P ) 2 = 2 ⌘ ( Q − P ) 2 + C, where is an integration constant. C

7 Exact analysis of a normal form system The form of solution : v ! ◆ 1 / 2 u A 1 / 2 ✓ t 1 1 A u q = ± ✏ + ⌘ sin ✓ ( t ) + cos ✓ ( t ) − , 2 − ⌘ ( ✏ + ⌘ ) ✏ + ⌘ v ! ◆ 1 / 2 u A 1 / 2 ✓ t 1 1 A u p = ± ✏ + ⌘ sin ✓ ( t ) − cos ✓ ( t ) − . 2 − ⌘ ( ✏ + ⌘ ) ✏ + ⌘ ✓ ↵ + � sn 2 ( t, k ) ◆ ⌘ ✓ ( t ) = arcsin . � sn 2 ( t, k ) − � − ✏ A 1 / 2 A := 1 + ( ✏ + ⌘ ) C sn( t, k ) : Jacobi elliptic sn function

8 Singularity structure(Riemann sheet) × , × : divergence point of q ( t ) ● : zero point of q ( t ) T : cut Im T 2 iK’ × × × × ● ● iK’ × × × × 0 ● ● 2 K 4 K 8 K 6 K Re T Time plane of q ( t ) K and K’ are the periods of sn function.

9 Topology of trajectory (single island chain case) × : divergence point of q ( t ) p ● : zero point of q ( t ) T : cut Im T 2 iK’ ● ● q iK’ ● ● × × ● 0 ● 2 K 4 K Re T Phase space with ✏ = ⌘ = − 2. Time plane of q ( t )

10 Topology of trajectory (single island chain case) × : divergence point of q ( t ) p ● : zero point of q ( t ) T : cut Im T 2 iK’ ● q iK’ × × 0 0 ● 2 K 4 K Re T Phase space with ✏ = ⌘ = − 2. Time plane of q ( t )

11 Topology of trajectory (single island chain case) × , × : divergence point of q ( t ) p ● : zero point of q ( t ) T : cut Im T 2 iK’ × × ● q iK’ × × 0 0 ● 2 K 4 K Re T Phase space with ✏ = ⌘ = − 2. Time plane of q ( t )

12 Topology of trajectory (single island chain case) × , × : divergence point of q ( t ) p ● : zero point of q ( t ) T : cut Im T 2 iK’ × × ● q iK’ × × 0 0 ● 2 K 4 K Re T Phase space with ✏ = ⌘ = − 2. Time plane of q ( t ) Imaginary actions for these topologically distinct paths are different.

13 Topology of trajectory (single island chain case) × , × : divergence point of q ( t ) p ● : zero point of q ( t ) T : cut Im T 2 iK’ × × ● q iK’ × × 0 0 ● 2 K 4 K Re T Phase space with ✏ = ⌘ = − 2. Time plane of q ( t ) Which is cheaper? In this case, the green one is the cheaper.

14 Topology of trajectory(double island chain case) H = 1 2( q 2 + p 2 ) + ✏ 4( q 2 + p 2 ) 2 + � 8 ( q 2 + p 2 ) 3 + ⌘ q 2 p 2 + ! q 4 p 4 × : divergence point of q ( t ) ● : zero point of q ( t ) T Im T : cut ● ● ● ● × × × ● Time plane of q ( t ) Re T Phase space with ✏ = − 2 , ⌘ = − 2 . 7 , � = 0 . 9 , ! = 1 . 8.

15 Topology of trajectory(double island chain case) H = 1 2( q 2 + p 2 ) + ✏ 4( q 2 + p 2 ) 2 + � 8 ( q 2 + p 2 ) 3 + ⌘ q 2 p 2 + ! q 4 p 4 × : divergence point of q ( t ) ● : zero point of q ( t ) T Im T : cut ● ● ● ● × × × ● Time plane of q ( t ) Re T Phase space with ✏ = − 2 , ⌘ = − 2 . 7 , � = 0 . 9 , ! = 1 . 8.

16 Topology of trajectory(double island chain case) H = 1 2( q 2 + p 2 ) + ✏ 4( q 2 + p 2 ) 2 + � 8 ( q 2 + p 2 ) 3 + ⌘ q 2 p 2 + ! q 4 p 4 × × : divergence point of q ( t ) ● : zero point of q ( t ) T Im T : cut ● ● × × × × × × ● ● × × × ● Time plane of q ( t ) Re T Phase space with ✏ = − 2 , ⌘ = − 2 . 7 , � = 0 . 9 , ! = 1 . 8.

17 Topology of trajectory(double island chain case) H = 1 2( q 2 + p 2 ) + ✏ 4( q 2 + p 2 ) 2 + � 8 ( q 2 + p 2 ) 3 + ⌘ q 2 p 2 + ! q 4 p 4 × × × : divergence point of q ( t ) ● : zero point of q ( t ) T Im T : cut × × × × × × ● ● × × × × × × ● ● × × × ● Time plane of q ( t ) Re T Phase space with ✏ = − 2 , ⌘ = − 2 . 7 , � = 0 . 9 , ! = 1 . 8. 3 possible imaginary actions.

18 ＋ Relation to the doublet case 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. ＝

19 Topology of trajectory(double island chain case) H = 1 2( q 2 + p 2 ) + ✏ 4( q 2 + p 2 ) 2 + � 8 ( q 2 + p 2 ) 3 + ⌘ q 2 p 2 + ! q 4 p 4 × × × : divergence point of q ( t ) ● : zero point of q ( t ) T Im T : cut × × × × × × ● ● × × × × × × ● ● × × × ● Time plane of q ( t ) Re T Phase space with ✏ = − 2 , ⌘ = − 2 . 7 , � = 0 . 9 , ! = 1 . 8. Looks different but the same topology, so the same imaginary action.

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. Two origins of the paths with different imaginary actions: 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" … ) 2. resolution of degenerated paths due to symmetry breaking.