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

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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

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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

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Simultaneous occurrence of direct and resonance

scattering mechanisms?

Simultaneous occurrence of direct and resonance

scattering mechanisms?

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Question:Question:

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

• Mathematical answer: yes by complex scaling transformation.

• Physical answer: ?

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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

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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

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Outgoing condition for resonances and CS

Outgoing condition for resonances and CS

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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

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States obtained by CS as scattering states for varying

projectile flux

States obtained by CS as scattering states for varying

projectile flux

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• Connection between gamma and theta:

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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

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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

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Wavefunctions of rotated continuum

Wavefunctions of rotated continuum

• exponentially modulated plane waves:

grows in x

decays in time

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• time-dependence:

2

2

exp

ˆexp exp

exp

i

i

i i

ip xe

i it Ht E e t

ip xe E te

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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 :

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

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

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Classical density for free particles:

Classical density for free particles:

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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

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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.

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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

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Construction of classical phase space density

Construction of classical phase space density

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

• phase space density:

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Construction of semiclassical wavefunction

Construction of semiclassical wavefunction

• dividing to incoming and outgoing parts:

• transformation of density to wf:

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The expected quantum counterpart

The expected quantum counterpart

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

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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:

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Comparison of scattering wavefunction and rotated

continuum state:

Comparison of scattering wavefunction and rotated

continuum state:

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Case III: near resonance scattering

Case III: near resonance scattering

• Potential:

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

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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

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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:

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slow change of gammaslow change of gamma

Im E

Re E

-0.0034

-0.002

0.7126 0.716

-0.004

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Resonance hit:Resonance hit:

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Off-resonance:Off-resonance:

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Off-resonanceOff-resonance

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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?

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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

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Scattering matrix from simulations:

Scattering matrix from simulations:

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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.

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

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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