imaging molecules with strong laser fields: successes and surprises misha ivanov
TRANSCRIPT
Olga Smirnova, MBI Berlin
Ryan Murray,University of Waterloo
Serguei Patchkovskii,NRC, Ottawa
Michael Spanner,NRC Ottawa
A la carte
• Ionization in strong IR fields & electronic structure – the naïve view
• Tunnel ionization of small molecules revisited
Ionization in strong IR fields
CO2
Naïve view: Tunnel ionization should map out the orbital
Agrees with experiment for HCl, N2, O2 – but not CO2
-eF
Dyson orbital for the CO2+ground state
Fix the axes,
Rotate F
X2g
~
Ionization in strong IR fields: Reality check
X2g
~Dyson orbital
Tunneling theory – dotted. Max @ 28o Experiment: Max @ 45o
Ionization in strong IR fields: Reality check
Standard tunneling theory – blue dotted. Max @ 28o
TDSE at 800 nm – solid: Max @ 45o : Same as experiment
Is tunneling idea fundamentally flawed? Let us revisit the theory
X2g
~Dyson orbital
A la carte
• Ionization in strong IR fields & electronic structure – the naïve view
• Tunnel ionization of small molecules revisited
• The basics
• Reminder: 1D WKB and the role of polarization
• 3D WKB: fitting a round peg into a square hole
• Results
A la carte
• Ionization in strong IR fields & electronic structure – the naïve view
• Tunnel ionization of small molecules revisited
• The basics
• Reminder: 1D WKB and the role of polarization
• 3D WKB: fitting a round peg into a square hole
• Results
1. Match polarized bound state at some plane z=z0 near the core
2. Outgoing wave at z=+ infinity
Eb= - 2/2Z
Z0
Outgoing wave
Polarized bound state
Tunnel Ionization as a boundary value problem
Tunnel Ionization Rate
Ionization rate: current across a plane z=z*
Eb= - 2/2Z
Z*
The role of the bound wavefunction is taken by the Dyson orbital
Our approach to single channel ionization
matching point
QuantumChemistry, polarized
3D WKB tunneling
How to approach 3D tunneling in a simple way?
A la carte
• Ionization in strong IR fields & electronic structure – the naïve view
• Tunnel ionization of small molecules revisited
• The basics
• Reminder: 1D WKB and the role of polarization
• 3D WKB: fitting a round peg into a square hole
• Results
Simple Example: 1D tunneling, I
Eb= - 2/2Z
Z0
Outgoing wave
We need to solve
subject to the boundary conditions at z0 and at +infinity
WKB
Simple Example: 1D tunneling, II
The WKB solution that matches the boundary conditions is
Eb= - 2/2Z
Z0
Outgoing wave
where s(z,z0) has to satisfy the HJ equation and the condition at z0
WKB
Simple Example: 1D tunneling, III
The solution of the HJ equation with this condition is
Eb= - 2/2Z
Z0
Outgoing wave
where
is velocity
WKB
Role of polarization, I
Eb= - 2/2
Z
Z0
Zex
Options:
•Correct: use polarized state – quantum chemistry for real systems
• Cheap: neglect polarization & use field-free bound state (??)
Need to know polarized bound state!
Eb= - 2/2
Z
Z0
Zex
Polarization appears in the exponent!
Field included
Field NOT included
Integrands are different, z0 does not cancel
Role of polarization, II
Errors here
Eb= - 2/2
Z
Z0
Zex
The trick: Expand exponent in Fz0/2 & keep only linear term.
Neglecting polarization & getting away with it
Effect scales with z02
… can only be done analytically
Example: short-range potentialExample: short-range potential
If U(z>z0) is small compared to Ip=2/2
Long-range potentials
If one does not include polarization, then one should use
short-range partshort-range part long-range correctionlong-range correction
A la carte
• Ionization in strong IR fields & electronic structure – the naïve view
• Tunnel ionization of small molecules revisited
• The basics
• Reminder: 1D WKB and the role of polarization
• 3D WKB: fitting a round peg into a square hole
• Results
WKB in 3D?
WKB in 3D is always a problem. What can we do here?
• P ≠0 puts exponential penalty on the rate, TUN is small
Experiment: W(P) ~ exp[-P2/P0
2]
• For z>>1 U(r)-Fz is almost separable in z and But the orbital But the orbital
is notis not
TUN
z, F
p≠
How does one force separability onto an arbitrary b?
Example: the short-range potential
The equation we need to solve
… is separable in x,y,z. But our boundary condition b is not.
How does one force separability onto an arbitrary b?
Eb= - 2/2
Z
Z0
Zex
Forcing separability: Partial Fourier transform
NB: The SE is linear
Use 2D Fourier in x,y to re-write b as sum of separable functions
Step 1. Solve the SE for each
with the boundary condition
),,(),,( 00 zppzpp yxbyx
Forcing separability: Partial Fourier transform
Step 2: Full solution is
The ionization rate can be calculated directly in the mixed space
Using Partial Fourier transform, I
The Schrödinger Equation for
yields effectively 1D equation for
The only difference from before is higher Ip for tunneling with px, py≠0
Using Partial Fourier transform, II
Then the answer is already known:
where the tunneling amplitude aT depends on px,py via effective Ip
tunneling time
Using Partial Fourier transform, III
The wavefunction just after the barrier is
And the ionization rate becomes
Long-range correction: modify v(z’) to include the core potential
Results and Analysis
Both coordinate and momentum matter!
In weak fields exp(-P2perpT) filtering is severe: T=[2Ip]1/2/F
Tunneling is along the field, The rate is coordinate-domain dominated
Z0
Analysis
Both coordinate and momentum matter!
2. For strong fields, filtering is not as severe – momentum features will start to show up
Predictions
momentumcoordinate
Weak fields: rate follows coordinate
Strong fields: rate mixes upcoordinate and momentum
HOMO (X-channel)
Results for N2
HOMO-1 (A-channel)
Angle deg
Ioniz
ati
on p
robabili
ty
I=8.1013 W/cm2
HOMO (X-channel)
HOMO-1 (A-channel)
I=1.7.1014 W/cm2
Angle deg
The matching point and polarized wavefunction
matching point
QuantumChemistry, polarized
3D WKB tunneling
If we implement our approach numerically, polarized wf must be used. Then z0 should drop out
Limiting cases
(L)=s,Atom R(L) Neglect the deviation angle T
The orbital density is imaged
directly
T
Limiting cases
(L)=s,Atom R(L) What about nodal planes?
Gives the PPT / Smirnov-Chibisov
result for atoms
Is equivalent to MO-ADK for
molecules
F0 is the orbital,
F1 its derivative vs
T
Conclusions
• Physically transparent theory for single – channel ionization in molecules
• Ionization images Dyson molecular orbitals in coordinate space if the fields are not too high.
• Close to barrier suppression intensities the momentum space features of the orbital become very important.
• CO2 experiments can be explained within the tunneling picture and a single-channel approximation
• Need to extend beyond barrier suppression and to oscillating fields