Quantum Mechanics of Mixed Systems Diego A. Wisniacki - PowerPoint PPT Presentation

Quantum Mechanics of Mixed Systems Diego A. Wisniacki UBA-Argentina Luchon 2015

Let us start with an example … ..

H ( p, q, t ) = 1 2 π cos(2 π p ) + k X 2 π cos(2 π q ) δ ( t − nk ) n p n +1 = p n + k sin(2 π q n ) (mod 1) , q n +1 = q n − k sin(2 π p n +1 ) (mod 1) ,

k=0.11 k=0.06 k=0.21 1 1 p 0 0 q 1 k=0.46 k=0.26 k=0.36

This transition is dictated by the KAM and the Poincare-Birkhoff theorems

I am interested the quantum manifestation of this transition

In the 80’ Uzer, Noid, Marcus JCP 83 Resonances AC Ozorio de Almeida JPC 84 In the 00’ Brodier, Schlagheck, Ullmo PRL’ 01 Resonances Tunneling Annals of Phys. ‘02 Eltschka, Schlagheck PRL ‘05 Lock, Backer, Ketzmerik, Schlagheck PRL ‘10 Hanada, Shudo, Ikeda arXiv ‘15

Quantum Harper map Map in a unit cell 1 1 p ~ = 2 π N 0 q 0 1 Evolution operator ˆ U k = exp[i Nk cos(2 π ˆ q )] exp[i Nk cos(2 π ˆ p )]

Quantum Harper map: spectra N=60 φ k

Quantum Harper map: eigenfunctions Husimi distribution H φ ( q, p ) = | h q, p | φ i | 2 • Positive real function • For maps in a torus has exact N zeros • The set of zeros of H encodes the full quantum information of the state

Quantum Harper map: eigenfunctions

Quantum Harper map: eigenfunctions

I want to study the ACs generated by a resonance

N=60 3 2 1 φ 0 -1 -2 -3 0.05 0.1 0.15 0.2 0.25 0.3 k n 1 =5 n 2 =11

3 2 1 φ 0 -1 -2 -3 0.05 0.1 0.15 0.2 0.25 0.3 k Δ n=10 Δ n=6

N 1 =5 n 2 =11 a (a) (c) (b) (d) (e) 1.2 f b φ gh c (f) (g) (h) (i) (j) d i e j -0.5 k 0.185 0.21

a (a) (c) (b) (d) (e) 1.2 f b φ gh c (f) (g) (h) (i) (j) d i e j -0.5 k 0.185 0.21

a (a) (c) (b) (d) (e) 1.2 f b φ gh c (f) (g) (h) (i) (j) d i e j -0.5 k 0.185 0.21

a (a) (c) (b) (d) (e) 1.2 f b φ gh c (f) (g) (h) (i) (j) d i e j -0.5 k 0.185 0.21

a (a) (c) (b) (d) (e) 1.2 f b φ gh c (f) (g) (h) (i) (j) d i e j -0.5 k 0.185 0.21

a (a) (c) (b) (d) (e) 1.2 f b φ gh c (f) (g) (h) (i) (j) d i e j -0.5 k 0.185 0.21

N=300 N=160 n 1 =9 n 2 =15 n 1 =18 n 2 =24 Δ n=6

N=300 n 1 =15 n 2 =27 n 1 =26 n 2 =44 Δ n=6*3=18 Δ n=6*2=12

Classical Quantum State localized in the islands r ceros r*(l-1) ceros Resonance r islands Δ n=r*l AC r*l ceros State localized in PO (Scaring???)

Semiclassical analysis of the gaps

h T n 1 | H | T n 2 i

Semiclassical analysis of the gaps ∞ X H r : s ' H 0 ( I r : s ) + V r,l ( I r : s ) cos( lr θ + φ l ) l =1 h n | H res | n + rl i = V r,l sc Z 2 π sc = e − i ϕ l V r,l exp( − irl θ ) δ I r : s ( θ ) d θ i π rlk 0 δ I r : s ( θ ) = I ( − 1) ( I r : s , θ ) − I r : s , ∆ φ ≈ | V r,l sc k | ~

Semiclassical analysis of the gaps h ∆φ Δ n=6 10 -4 10 -4 10 -5 h ∆φ Δ n=10 10 -6 10 -8 k 0.11 0.21 10 -7 0.16 0.21 0.26 0.31 k r=10 r=6

Semiclassical analysis of the gaps r=6 − h ∆φ 10 -6 Δ n=6 (a) 10 -9 N=80 10 -4 N=300 1 ~ = − h 2 π N ∆φ Δ n=6*2 N=160 (b) 10 -11 10 -5 − h ∆φ Δ n=6*3 10 -12 (c) 0.16 0.18 0.2 0.22 0.24 0.26 k

Semiclassical analysis of the gaps 10 -8 − ∆φ 3 h 10 -18 5 h ∆φ − 10 -28 0.16 0.19 0.22 0.25 k

Universality Li LiCN-LiNC C N

Applications

Experimental observation

• Classical nonlinear resonance imprints a systematic influence in the quantum eigenvalues and eigenfunctions of a mixed system. • Universal structure embedded in the spectra: states localized in tori interact if the quantum numbers differ in a multiple of the order of the resonance. • Eigenstates in the AC has a particular morphology. One state is localized in the vicinity of the unstable PO and the other state is localized on the island chain. • These findings could be of importance in the design of optical microcavities.