Resonances and background scattering in gedanken experiment with varying projectile flux

Petra Zdanska, IOCBPetra Zdanska, IOCB

June 2004 – Feb 2006June 2004 – Feb 2006

Resonances and Resonances and background scattering in background scattering in

gedanken experiment with gedanken experiment with varying projectile fluxvarying projectile flux

IOCB IOCB February 20, 200February 20, 20066

22

Personal acknowledgement

Personal acknowledgement

• Milan Sindelka and Nimrod Moiseyev

• Vlada Sychrovsky and people attending my unfinished Summer course of resonances 2004

• Nimrod’s group and conferences


33

Resonance and direct scattering as two mechanisms

Resonance and direct scattering as two mechanisms

• Direct– density of states

changes evenly smooth spectrum

• Resonance– metastable states– density of states

includes peaks


44

Simultaneous occurrence of direct and resonance

scattering mechanisms?

Simultaneous occurrence of direct and resonance

scattering mechanisms?


55

Question:Question:

• Are direct and resonance scattering mechanisms separable at near resonance energy ?

• Mathematical answer: yes by complex scaling transformation.

• Physical answer: ?


66

Complex scaling method (CS)

Complex scaling method (CS)

• useful non-hermitian states – “resonance poles”– purely outgoing condition is a cause to

exponential divergence and complex energy eigenvalue

• complex scaling transformation of Hamiltonian– non-unitary similarity transformation for

taming diverging states


77

exp exp cos Re sin Im

exp sin Re cos Im

Imarctan arctan

Re Re

i

c

ipxe ix p p

x p p

p

p p

Ougoing condition for resonances and CS

Ougoing condition for resonances and CS

• Problem:• Solution:

exp exp Re Imipx ix p x p


88

Outgoing condition for resonances and CS

Outgoing condition for resonances and CS


99

Separation of direct and resonance scattering by CS

transformation

Separation of direct and resonance scattering by CS

transformation

Im E

Re Eboundstates resonance

rotated continuum


1010

States obtained by CS as scattering states for varying

projectile flux

States obtained by CS as scattering states for varying

projectile flux


1111

• Connection between gamma and theta:


1212

Proofs by semiclassical and quantum simulations

Proofs by semiclassical and quantum simulations

• Why semiclassical and not just quantum mechanics – only way to prove a correspondence between the

classical notion of flux of particles and quantum wavefunctions

• Cases I and II:– I. analytical proof for free-particle scattering– II. numerical evidence for direct scattering problem

• Case III:– a quantum simulation of resonance scattering for

varying projectile flux displaying the new effects


1313

Case I: Free-particle Hamiltonian

Case I: Free-particle Hamiltonian

• non-hermitian solutions of CS Hamiltonian:

2

Im E

Re E

2 22

2

ˆ ˆˆ ˆ2 2

ˆ

exp exp

i

i

i

p pH H e

H

E e

ipx ipxe


1414

Wavefunctions of rotated continuum

Wavefunctions of rotated continuum

• exponentially modulated plane waves:

grows in x

decays in time


1515

• time-dependence:

2

2

exp

ˆexp exp

exp

i

i

i i

ip xe

i it Ht E e t

ip xe E te


1616

Semiclassical solution to the expected physical process

behind these non-hermitian states:

Semiclassical solution to the expected physical process

behind these non-hermitian states:

• step I: construction of a corresponding density probability in classical phase space– 1st order emission in an asymptotic

distance xe with the rate :


1717

– density of particles in a close neighborhood of the emitter:

– analytical integration of the classical Liouville equation with the above boundary condition:


1818

Classical density for free particles:

Classical density for free particles:


1919

Step II: transformation of classical phase space density to a quantum wavefunction

Step II: transformation of classical phase space density to a quantum wavefunction– non-approximate, in the case of free-

Hamiltonian


2020


2121

Exact comparison with non-hermitian wavefunction as a

proof

Exact comparison with non-hermitian wavefunction as a

proof• the non-hermitian and scattering

wavefunctions have the same form and are equivalent supposed that,

– which was to be proven.


2222

Case II: Rotated complex continuum of Morse oscillator

Case II: Rotated complex continuum of Morse oscillator

• potential:

• semiclassical simulation of scattering experiment with parameters:– particles arrive with classical energy:– decay rate of the emitter:

11 . ., 1 . . , 10 . .D a u a u a u


2323

Construction of classical phase space density

Construction of classical phase space density

• classical orbit [x(t),p(t)] is evaluated

• phase space density:


2424

Construction of semiclassical wavefunction

Construction of semiclassical wavefunction

• dividing to incoming and outgoing parts:

• transformation of density to wf:


2525


2626


2727

The expected quantum counterpart

The expected quantum counterpart

• Non-hermitian solution of CS Hamiltonian with the energy:


2828

Solution of CS Hamiltonian in finite box:

Solution of CS Hamiltonian in finite box:

• box: • N=200 basis functions• solution of CS Hamiltonian:

• back scaled solution:


2929

Comparison of scattering wavefunction and rotated

continuum state:

Comparison of scattering wavefunction and rotated

continuum state:


3030

Case III: near resonance scattering

Case III: near resonance scattering

• Potential:

• Examined scattering energies:– resonance hit– very slightly off-resonance


3131

in complex energy plane:in complex energy plane:

Im E

Re E

-0.0034

-0.002

V(x)

x

0.7126 0.716

-0.004


3232

Quantum dynamical simulations of scattering

experiments

Quantum dynamical simulations of scattering

experiments• “particles” added as Gaussian

wavepackets in an asymptotic distance, 40 a.u.

• beginning of simulation: scattering experiment does not start abruptly but the intensity I(t) is modulated as follows:


3333

slow change of gammaslow change of gamma

Im E

Re E

-0.0034

-0.002

0.7126 0.716

-0.004


3434

Resonance hit:Resonance hit:


3535

Off-resonance:Off-resonance:


3636

Off-resonanceOff-resonance


3737

What is going on:What is going on:

• We reach stationary-like scattering states, which are characterized by a constant scattering matrix and by a constant (and complex) expectation energy value.

• Are these states the non-hermitian solutions to Hamiltonian obtained by CS method?


3838

Calculations of scattering matrix:

Calculations of scattering matrix:

• comparison of dynamical simulations with stationary solutions of complex scaled Hamiltonian

• gamma<Gamma_res :– rotated continuum

• gamma>Gamma_res :– resonance hit resonance pole– slightly off-resonance rotated

continuum


3939

Scattering matrix from simulations:

Scattering matrix from simulations:


4040

Inverted control over dynamics for gamma>Gamma_res

Inverted control over dynamics for gamma>Gamma_res

• incoming flux decays faster than the wavefunction trapped in resonance

• natural control: incoming flux disappears faster than outgoing flux – this occurs for discrete resonance energies

• inverted control: outgoing flux decays according to gamma and not Gamma_res. Reason: destructive quantum interference removes the trapped particle.


4141

• empirical rule in CS: rotated continuum for θ> θc (γ>Γres) is not responsible for resonance cross-sections.


4242

Conclusions:Conclusions:

• resonance phenomenon studied in a new context of scattering dynamics

• new light shed into complex scaling method, interference effect behind the long accepted empirical rule

• first physical realization of complex scaling eventually interesting for experiment

Resonances and background scattering in gedanken experiment with varying projectile flux

Documents

Transcript of Resonances and background scattering in gedanken experiment with varying projectile flux