Transcript Resonance hit

Resonances and background
scattering in gedanken
experiment with varying
projectile flux
Petra Zdanska, IOCB
June 2004 – Feb 2006
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
• Direct
– density of states
changes evenly 
smooth spectrum
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• Resonance
– metastable states
– density of states
includes peaks
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Simultaneous occurrence of direct
and resonance scattering
mechanisms?
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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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?
5
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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Ougoing condition for resonances
and CS
• Problem: exp  ipx   exp  ix Re p  x Im p 
• Solution:
exp  ipxe
i
  exp ix  cos Re p  sin  Im p 
 exp   x  sin  Re p  cos  Im p  
 Im p

 c  arctan
 arctan
Re p
Re p
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Outgoing condition for
resonances and CS
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Separation of direct and
resonance scattering by CS
transformation
Re E
bound
states
resonance
rotated continuum
Im E
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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
• 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
• non-hermitian solutions of CS
Hamiltonian:
Re E
2
2
2
ˆ
ˆ
p
p
Hˆ 
 Hˆ  
e 2i
2
2
Hˆ    

Im E
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

   E e 2i
   exp  ipx     exp  ipxe  i 
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Wavefunctions of rotated
continuum
• exponentially modulated plane waves:
decays in time
grows in x
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• time-dependence:
i
 i 
  exp  p xe 


 i
 i ˆ 
2 i 
  t   exp   Ht    exp   E e t  




i
2 i 
i
 exp   p xe  E te  


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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:
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Step II: transformation of classical
phase space density to a quantum
wavefunction
– non-approximate, in the case of freeHamiltonian
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
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Exact comparison with nonhermitian 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
• potential:
D  1a.u.,   1a.u.1 ,   10a.u.
• semiclassical simulation of scattering
experiment with parameters:
– particles arrive with classical energy:
– decay rate of the emitter:
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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
• dividing to incoming and outgoing
parts:
• transformation of density to wf:
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The expected quantum counterpart
• Non-hermitian solution of CS
Hamiltonian with the energy:
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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:
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Case III: near resonance scattering
• Potential:
• Examined scattering energies:
– resonance hit
– very slightly off-resonance
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in complex energy plane:
V(x)
0.7126
0.716
Re E
-0.002
-0.0034
-0.004
Im E
x
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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 gamma
0.7126
0.716
Re E
-0.002
-0.0034
-0.004
Im E
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Resonance hit:
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Off-resonance:
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Off-resonance
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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:
• 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:
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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:
• 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
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