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Quantum Mechanics of Mixed Systems Diego A. Wisniacki - - PowerPoint PPT Presentation
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 cos(2 p ) + k X 2 cos(2 q ) ( t nk ) n p n +1 = p n + k sin(2 q n ) (mod 1) , q n +1
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pn+1 = pn + k sin(2πqn) (mod 1), qn+1 = qn − k sin(2πpn+1) (mod 1),
H(p, q, t) = 1 2π cos(2πp) + k 2π cos(2πq) X
n
δ(t − nk)
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q p 1 1 k=0.06 k=0.11 1 k=0.21 k=0.26 k=0.36 k=0.46
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This transition is dictated by the KAM and the Poincare-Birkhoff theorems
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I am interested the quantum manifestation of this transition
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In the 80’ Resonances AC
Uzer, Noid, Marcus JCP 83 Ozorio de Almeida JPC 84
In the 00’ Resonances Tunneling
Brodier, Schlagheck, Ullmo PRL’ 01 Annals of Phys. ‘02 Eltschka, Schlagheck PRL ‘05 Lock, Backer, Ketzmerik, Schlagheck PRL ‘10 Hanada, Shudo, Ikeda arXiv ‘15
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Quantum Harper map
ˆ Uk = exp[iNk cos(2πˆ q)] exp[iNk cos(2πˆ p)]
~ = 1 2πN
Evolution operator Map in a unit cell
1 1 q p
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Quantum Harper map: spectra
k φ
N=60
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Quantum Harper map: eigenfunctions
Hφ(q, p) = |hq, p|φi|2
Husimi distribution
- 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
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Quantum Harper map: eigenfunctions
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Quantum Harper map: eigenfunctions
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I want to study the ACs generated by a resonance
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k φ
- 3
- 2
- 1
1 2 3 0.05 0.1 0.15 0.2 0.25 0.3
n1=5 n2=11
N=60
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- 3
- 2
- 1
1 2 3 0.05 0.1 0.15 0.2 0.25 0.3
k φ Δn=6 Δn=10
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- 0.5
1.2 0.185 0.21
k φ
f gh c d i j e (f) (g) (h) (i) (j) (e) (d) (b) (a) (c) a b
N1=5 n2=11
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- 0.5
1.2 0.185 0.21
k φ
f gh c d i j e (f) (g) (h) (i) (j) (e) (d) (b) (a) (c) a b
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- 0.5
1.2 0.185 0.21
k φ
f gh c d i j e (f) (g) (h) (i) (j) (e) (d) (b) (a) (c) a b
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- 0.5
1.2 0.185 0.21
k φ
f gh c d i j e (f) (g) (h) (i) (j) (e) (d) (b) (a) (c) a b
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- 0.5
1.2 0.185 0.21
k φ
f gh c d i j e (f) (g) (h) (i) (j) (e) (d) (b) (a) (c) a b
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- 0.5
1.2 0.185 0.21
k φ
f gh c d i j e (f) (g) (h) (i) (j) (e) (d) (b) (a) (c) a b
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N=300 N=160 n1=9 n2=15 Δn=6 n1=18 n2=24
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N=300 n1=15 n2=27 n1=26 n2=44 Δn=6*2=12 Δn=6*3=18
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Resonance r islands
Δn=r*l
AC Classical Quantum
r ceros r*(l-1) ceros r*l ceros
State localized in the islands State localized in PO (Scaring???)
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Semiclassical analysis of the gaps
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hTn1| H |Tn2i
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Semiclassical analysis of the gaps
Hr:s ' H0(Ir:s) +
∞
X
l=1
Vr,l(Ir:s) cos(lrθ + φl)
δIr:s(θ) = I(−1)(Ir:s, θ) − Ir:s,
V r,l
sc = e−iϕl
iπrlk Z 2π exp(−irlθ) δIr:s(θ)dθ
∆φ ≈ |V r,l
sc k|
~ hn| Hres |n + rli = V r,l
sc
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Semiclassical analysis of the gaps
10-7 10-6 10-5 10-4 0.16 0.21 0.26 0.31
10-8 10-4 0.11 0.21
h ∆φ
h∆φ
k
k
Δn=6 Δn=10
r=6 r=10
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Semiclassical analysis of the gaps
Δn=6 Δn=6*2 Δn=6*3
k
−
h ∆φ
−
h ∆φ
−
h
10-12 10-5 0.16 0.18 0.2 0.22 0.24 0.26 10-11 10-4 10-9 10-6
(a) (c) (b)
∆φ
r=6
~ = 1 2πN
N=80 N=300 N=160
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Semiclassical analysis of the gaps
k
∆φ
5
h
10-28 10-18 10-8 0.16 0.19 0.22 0.25
− ∆φ
3
h
−
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Universality
LiCN-LiNC C N Li
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Applications
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Experimental observation
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- 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
- f 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.
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