Lifetime distributions in open quantum systems: beyond ballistic - - PowerPoint PPT Presentation

▶

Feb 24, 2023 643 likes •848 views

Lifetime distributions in open quantum systems: beyond ballistic chaotic decay Henning Schomerus Lancaster University CIRM, 22 January 2009 With J Tworzyd o, M Kopp, J Wiersig, J Main, J Keating, M Novaes Stroboscopic scattering theory: round

SLIDE 1

Lifetime distributions in open quantum systems: beyond ballistic chaotic decay

Henning Schomerus Lancaster University CIRM, 22 January 2009

With J Tworzydło, M Kopp, J Wiersig, J Main, J Keating, M Novaes

SLIDE 2

exit: PTFP PTF(QF)P PTF(QF)2P PTF(QF)3P PTF(QF)4P… inject a particle:

Resonances:

2 / ; ; Γ − = ≡ =

− −

i E e e QFQ

i i

ε λ ψ ψ

ε ε

( ) FP

FQ e P S

i T 1

) ( matrix S FT

− − −

= ⇒

ε

Stroboscopic scattering theory:

round‐trip operator F, dim F=M= 1/h; opening operator P=(MxN) , internal space: projector Q=1‐PPT

SLIDE 3

exit: PTFP PTF(QF)P PTF(QF)2P PTF(QF)3P PTF(QF)4P… inject a particle:

Resonances:

2 / ; ; Γ − = ≡ =

− −

i E e e QFQ

i i

ε λ ψ ψ

ε ε

( ) FP

FQ e P S

i T 1

) ( matrix S FT

− − −

= ⇒

ε

Stroboscopic scattering theory:

round‐trip operator F, dim F=M= 1/h; opening operator P=(MxN) , internal space: projector Q=1‐PPT

SLIDE 4

exit: PTFP PTF(QF)P PTF(QF)2P PTF(QF)3P PTF(QF)4P… inject a particle:

Resonances:

2 / ; ; Γ − = ≡ =

− −

i E e e QFQ

i i

ε λ ψ ψ

ε ε

( ) FP

FQ e P S

i T 1

) ( matrix S FT

− − −

= ⇒

ε

Qm‐cl correspondence Goal: exploit this for resonance states

Stroboscopic scattering theory:

SLIDE 5

Challenge: quasi‐deterministic decay

Nominally diverging decay rates:
Resonance wave functions quasi‐degenerate

(defective eigensystem)

) (

1 1 lim

t t

e

− Γ ∞ → Γ

+ ) exp(Im | | = = ε λ

SLIDE 6

illustration: standard map/kicked rotator

) 1 (mod ) 2 sin( ) 1 (mod

1 1 1

+ + +

+ = + =

n n n n n n

x K p p p x x π

(classical) (qm)

)] 2 (cos ) ( exp[ 1

2 2 M m iMK M i nm

n m iM F π

π π

− − =

K=2 K=7.5 K=7.5, M=1280, N=256

λ ε

Resonances wave functions Escape zones

SLIDE 7

Classically chaotic systems (with J Tworzydło):

fractal Weyl law (see M Zworski)

– Goal: reinstate phase space rules

Mixed phase space (with M Kopp):

… fractal Weyl law …

– Goal: test character of chaotic component

Refractive escape (with J Wiersig; J Keating and M Novaes):

dielectric resonators

– Goal: generalization and comparison to realistic systems

SLIDE 8

Resonance distribution

Fractal Weyl law Power law scaling

Classically chaotic systems

SLIDE 9

Resonance distribution

Fractal Weyl law Power law scaling

Classically chaotic systems

SLIDE 10

A. identify short‐lived deterministic dynamics in phase space

=

QFQψ

) , ( ∞ = Γ =

n n

λ

Define P=PT P=1‐Q
trivially: QP=0 → N states on opening (Po=P)
semicl.: preimage: projector P1=P1P1

naïve Weyl: dim = area/Planck= M • area

problem: underestimates no. of states reason: operator not self‐adjoint, states nonorthog., highly degenerate

Try to count short‐living states

SLIDE 11

2nd preimage, projector P2=P2P2

3rd preimage, projector P3=P3P3

tth preimage, projector Pt=PtPt

semiclassical propagation:
B. Cure degeneracy

: consider

) (

) 1 (

= =

n n

QFQ λ ψ

) ( ) ( ) 1 ( ) 1 ( t n t n t n n t n

QFQ ψ ψ ψ λ ψ = + =

+ +

) ( , ) (

s t t

s t QFQ t ≠ = =

P P P

C. Requires: areas

Weyl:

Ehr

t M t M t A ≡ Λ < ⇒ > Λ − ≈ ) ln( 1 / 1 ) exp(

) 1 ( rank

/ t

dwell Ehr t

t Ehr

e M t t

−

− = <

∑

P

dwell dwell Ehr

t t t

M Me

Λ − −

∝

/ 1 1 /

D. Remaining states (long living):

SLIDE 12

What have we done? A semiclassical partial Schur decomposition!

Pt : part of orthogonal basis U in QFQ=UTU+

where T is triangular with evals on diagonal. Test: Husimi rep. of Schur vectors (|λn|<0.1, M=1280)

SLIDE 13

Mixed phase space

Position of leads is important; coupled islands: fast decay Uncoupled islands: slow tunneling escape

SLIDE 14

Two accumulation regions:

1 , | | ≈ λ

idea: fix both upper and

lower cut‐off of lifetimes

uncoupled islands

(long‐living states): just the ordinary Weyl law…

SLIDE 15

SLIDE 16

Slightly unexpected…

Time domain studies: classical part of mixed phase space is quite unlike a fully chaotic phase space: Power law decay vs exponential decay Origin: sticking to islands

(see eg Cristadoro/Ketzmerick PRL 08)

Possible explanations: a)The fractal Weyl law actually breaks down for much larger M b)Sticking just contributes to the long‐living states c)Areas also power‐law distributed?

α −

∝ t ) / exp(

dwell

t t − ∝

SLIDE 17

Generalization: nonballistic escape

Applications: q‐dots w/tunnel barriers, dielectric resonators Stroboscopic scattering operator For dielectric resonators: with frequency ω, traversal time τ = n π A / v C (Sabine’s law), and R, T determined by Fresnel reflection coefficients. (n: refractive index; A: area, C: perimeter, v: velocity) Also, M=N=dim S= ω C/v π (Weyl’s law applied to the boundary)

( ) FT

FR e T R S

i 1

' ' ) (

− − −

+ =

ε

( ) FT

FR e T R S

i 1

) (

− −

− + − =

ωτ

ω

SLIDE 18

Compare realistic resonator to random matrix theory (RMT)

Bands of short‐living states (origin: bouncing ball motion) Requires to renormalize M and τ! Here done independent from fluctuations by using mean level spacing and decay rate of long‐living states.

SLIDE 19

Phase space rules can be resurrected by semiclassical Schur

decomposition; links fractal Weyl law to Ehrenfest time

Fractal Weyl law also exists in

generic dynamical systems (mixed phase space)

Stroboscopic scattering theory