AU
LTC
DEPARTMENT OF PHYSICS
AND ASTRONOMY
LUNDBECK FOUNDATION
THEORETICAL CENTER FOR
QUANTUM SYSTEM RESEARCH
FACULTY OF SCIENCE
AARHUS UNIVERSITY
LUNDBECKFONDEN
Monte Carlo Wave Packet
approach to Dissociative Multiple
Ionization in Diatomic Molecules
Henriette A. Leth, Lars Bojer Madsen, Klaus Mølmer
Outline
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• Introduction.
• The Monte Carlo Wave Packet (MCWP) method:
• Simple example.
• In general.
• Equivalence to the Master equation.
• Dissociative double ionization of H2.
• Results on H2.
• Outlook.
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Introduction
Dissociative Ionization
Short, Intense Laser Pulses
• Intensity:
• Wave length:
• Duration:
Extremely Difficult
• Many degrees of freedom.
• Both ionization and dissociation.
• Very strong coupling.
• Mixture of timescales.
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1-3×1014 W/cm2
480-1200 nm
40-140 fs
Free Electron Lasers
• Intensity:
• Wave length:
• Duration:
1-3×1013 W/cm2
5-50 nm
10-100 fs
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Introduction
Dissociative Ionization
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Monte Carlo Wave Packet
(MCWP)
• Stochastic technique from quantum
optics.
• Simulate the detection of continuum
electrons.
• Nuclear dynamics in every charge state.
Extremely Difficult
• Many degrees of freedom.
• Both ionization and dissociation.
• Very strong coupling.
• Mixture of timescales.
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Introduction
Dissociative Ionization
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Results at 1x1014 W/cm2
• Kinetic energy release spectra for the
two nuclei.
• Staudte et al., Phys. Rev. Lett. 98,
073003 (2007).
Extremely Difficult
• Many degrees of freedom.
• Both ionization and dissociation.
• Very strong coupling.
• Mixture of timescales.
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Dissipative Processes
Dissipative
Processes
The Master Equation
• Introduce the system density matrix
• Quantum system not isolated
from its surroundings.
• Dynamics beyond that
contained in the usual TDSE.
• Combine rate equations and coherent dynamics.
• The Master equation:
• E.g. spontaneous emission.
E
0
• Transition operator:
Isolated system
Relaxation term
1
• The relaxation term is on the general Lindblad
form:
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Dissipative Processes
Dissipative
Processes
• Quantum system not isolated
from its surroundings.
• Dynamics beyond that
contained in the usual TDSE.
Monte Carlo Wave Packet
• Simulate the Master equation.
• Apply wave functions rather than density matrices.
• Include a stochastic element.
• Include non-Hermitian Hamiltonian:
• E.g. spontaneous emission.
E
0
• Transition operator:
Isolated system
Relaxation terms
1
• Drop in norm in combination with random numbers
determine the instant of transition.
• Average over many evolutions.
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Calculational strategy
Monte Carlo Wave Packet
• Non-Hermitian Hamiltonian:
• Transition operator:
Strategy
• Many evolutions with different
transition times.
• Average over outcomes gives physical
correct result.
Summary
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Simple Example
Monte Carlo Wave Packet
• Non-Hermitian Hamiltonian:
• Transition operator:
Two Level System
Calculations
• Transition operator:
• Hamiltonian:
• Initial state:
E
• Time evolution operator:
0
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Simple Example
• Wave function at later time:
Calculations
• Transition operator:
• The norm squared (neglect dt2):
• Hamiltonian:
• Random between 0 and 1
• < dp • Transition
• > dp • No transition
• Renormalization
• Initial state:
• Time evolution operator:
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Simple Example
No transition or
renormalization
• Wave function at later time:
norm2
1
• The norm squared (neglect dt2):
0.5
0
• Random between 0 and 1
• Renormalization
25
50
time
Result of one run
• < dp • Transition
1
norm2
• > dp • No transition
0
0.5
0
0
25
time
50
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Simple Example
Two Level System
Input
Halftime
E
0
norm2
1 run
20 runs
1000 runs
1
1
1
0.5
0.5
0.5
0
0
0
25
time
50
0
0
25
time
50
0
25
time
50
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Calculational strategy
MCWP in General
• Non-Hermitian Hamiltonian:
No Jump (
> dP)
• Renormalize:
• Propagate using non-Hermitian H:
Jump (
< dP)
• Apply transition operator:
• Determine the reduction in norm2:
• Which one is determined from
branching ratios:
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Equivalence to Master Eq
Equivalence
• Consider the operator:
No Jump (
> dP)
• Renormalize:
• Time average (many realizations):
Renormalized
No-jump
wave function
probability
Jump (
< dP)
• Apply transition operator:
Jump
probability Probability to
jump using
Wave function
after jump
• Which one is determined from
branching ratios:
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Equivalence to Master Eq
Equivalence
• After reduction:
• Consider the operator:
• Time average (many realizations):
Renormalized
No-jump
wave function
probability
• The Hamiltonian:
• The wave function after
propagation using H:
Jump
probability Probability to
jump using
Wave function
after jump
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Equivalence to Master Eq
• Neglect terms including dt2:
• After reduction:
• The Hamiltonian:
• The wave function after
propagation using H:
• The Lindblad operator:
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Equivalence to Master Eq
• Neglect terms including dt2:
• Use the definition of (t):
• To obtain:
• Rearrange:
• The Lindblad operator:
• The Master equation:
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The Monte Carlo Method
Physical Interpretation
• MCWP can be obtained directly from
the system-environment problem
(alternative derivation of Master eq).
• The system evolves into a superposition
of different charge states.
• Simulate the effect of a measurements
and project the state according to the
random outcome.
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The Born-Markov
Approximation:
• The interaction between system and
environment is weak:
• Transition probability is small in
every timestep
• Environment is Markovian:
• Quantum excitations of the
environment degrees of freedom
spreads rapidly.
• Previous dissipated excitations are
not reabsorbed by the system.
• A measurement may not change
the future evolution.
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Double Ionization of H2
Double Ionization of H2
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How to apply to H2:
• Born-Oppenheimer approximation
• Input:
• Electronic structure
• Dipole moment functions
• Ionization rates
Basis
c
• Electronic solutions
• The total state
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Double Ionization of H2
Hamiltonian
The Hermitian part
• The total Hamilton
operator:
• The Hermitian part of
the Hamiltonian contains
three terms:
Jump Operators
• Three operators are included:
Nuclear
kinetic
energy
Electronic
kin and pot
Electron-field
coupling
c
u
g
h
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Double Ionization of H2
Potentials and Wave Functions
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Summary
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Important Points
Instant for the First
Ionization
• First ionization early in the pulse ->
• Time to dissociate before the second
ionization ->
• Larger internuclear separation at
second ionization ->
• Less potential energy.
Time for first
ionization
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Intensity Components
• Several different intensity maxima
depending on the position in the laser
beam.
• High intensity ->
• The first ionization appears earlier ->
• Focal volume averaging is important.
Peak intensity
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Results
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Physical Explanations
Dissociation goes
slower in D2 ->
more potential
energy
Charge resonance enhanced
ionization (CREI)
Long pulse-> more
dissociation
• Staudte et al., Phys. Rev. Lett. 98, 073003 (2007).
Characteristic
feature of the
dipole coupling.
Ionization late in
pulse -> no
dissociation
Kinetic energy
obtained prior to
ionization
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Oxygen in FEL
Ionization of Oxygen
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Born Oppenheimer
No cross sections
Free Electron Lasers
• Intensity:
• Wave length:
• Duration:
1-3×1013 W/cm2
5-50 nm
10-100 fs
Cross sections
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Conclusion
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• Dissociative multiple ionization as a systemenvironment problem.
• Solution equivalent to solving the Master equation,
however:
• Including wave functions.
• Stochastic jumps between pure states.
• Interpreted as measurements.
• Low computational cost.
• Agreement with experiments.
• Future applications
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Deleted Slides
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Double Ionization of H2
Simpler Approach
• The evolution is the same in every simulation up till
the first jump.
• Use this to do the calculation once and weight with
suitable probabilities.
• Norm squared if no renormalization:
• Jump probability at time T1:
• Do the same for the second ionization, given the first
ionization time.
• Total result:
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Summary
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Double Ionization of H2
The Hermitian part
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Evolution of the Nuclear Wave
Function
• The Schrödinger equation:
Nuclear
kinetic
energy
Electron- Electronic
field
kin and pot
coupling
Basis
• Insert the state ket and project onto
to obtain the evolution of
the nuclear wave
functions:
c
• Electronic solutions
g
• The total state
u
h
• Three terms have to be
evaluated.
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Double Ionization of H2
The Hermitian part
Nuclear
kinetic
energy
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H_elec
BO-potential
Electron- Electronic
field
kin and pot
coupling
L_elec
• Length gauge:
T_nuc
• Neglect rotations and use BornOppenheimer:
Molecular
orientation (fixed) Dipole moment
function
• Only coupling not included as jumps is the
1s g-2p u coupling.
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Double Ionization of H2
Determine the Hamiltonian
• We have:
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The State Ket
Nuclear wave function:
• Put in the results for the matrix elements, add a few
identities and rearrange to:
The reduced wave
function
• Where:
Tight static internuclear
orientation
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The Jump Operators
Born-Oppenheimer
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Jump Operators
c
u
g
Field-dressed states
h
• Alternatively one may use
• The KER spectra are almost identical
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The Jump Operators
Conclusion
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Jump Operators
• Total non-Hermitian Hamiltonian:
• With the Hermitian part:
Field-dressed states
• Alternatively one may use
• The KER spectra are almost identical
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Results
Laser Parameters
1×1014 W/cm2
• Wave length:
• Duration:
• Keldysh parameter:
• Norm squared nuclear wave function:
800 nm
40 fs
1.2-1.3
The Nuclear Wave
Function
nonnormalized
• Probability
of being in
the neutral
state:
2
• Intensity:
The Neutral Molecule
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Results
1s
g
• Equilibrium internuclear separation
larger than in H2 ->
• Oscillations (Tvib = 15 fs).
• Coupling to 2p u.
2p
u
• Antibonding curve -> Dissociation.
• Modulations due to oscillations in the
laser field.
3 photon resonance
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High Energy Peaks
Experiential Observations
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Explanation
• High energy peak growing with increasing field
strength.
• Three-photon coupling
in the 1sσg-2pσu system
• Litvinyuk et al, New. J. of Phys. 10, 083011 (2008).
• Ionization primarily from
2pσu
480 nm:
9 eV
600 nm:
7.5 eV