Imaging molecules with strong laser fields: successes and surprises Misha Ivanov

Imaging molecules with strong laser fields: successes and surprises

Misha Ivanov

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

Dyson orbital

X2g

~


X2g

~Dyson orbital

Tunneling theory – dotted. Max @ 28o Experiment: Max @ 45o


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

The Talk Trailer*

X2g~ Abu-samha &

Madsen, PRA 2009

*: approved for all audiences

Our results

A la carte



• The basics

• Reminder: 1D WKB and the role of polarization

• 3D WKB: fitting a round peg into a square hole

• Results

A la carte



• The basics



• 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



• The basics



• 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

Result in 1D

Eb= - 2/2Z

Z0Zex

z> zex

Need to know polarized bound state!

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



• The basics



• 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

Example: CO2

X2g~

Abu-samha &Madsen, PRA 2009

Results for C02

TDSE, numerics for 1600 nm

3D WKB, analytics

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

Test with H2

It works! Ratio of 0 to 90 deg is 1.4

Simple analytical results

Let

Then (L)=s,Atom R(L)

T

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? F0=0!!

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

Full expression, strong fields

(L)=s,Atom R(L)

R=

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

Tunnel Ionization as a boundary value problem

with boundary with boundary conditions:conditions:

•Stationary SE (SSE)

1. Match to a polarized bound state at some plane z=z0 near the core

2. Outgoing wave at z=+ infinity

Documents

Imaging molecules with strong laser fields: successes and surprises Misha Ivanov