petra zdanska, iocb june 2004 – feb 2006 resonances and background scattering in gedanken...
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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