Contributions to the Theory ofAtoms and Molecules in Strong

Electromagnetic Fields

Defence of Dissertation for the Degree of PhD.

Sølve Selstø

Department of Physics and TechnologyUniversity of Bergen

The atom

Light

The interaction between atoms and light

Hydrogen atoms in strong, short laser pulses

The hydrogen molecular ion in laser pulses

The Landau-Zener model and single electron transport in adouble quantum dot

Summary

Atoms

http://www.physics.queensu.ca/˜nanophys/stm.html

9 nm × 6.3 nm

Some ”highlights” in the evolution of quantum mechanics

Planck and Einstein, early 1900-s: Lightsometime behaves as particles, not just waves

Bohr-model, 1913

DeBroglie, 1923: Matter sometimes behave aswaves, not just particles

Schrodinger, 1932: Wave description of matter

Some ”highlights” in the evolution of quantum mechanics

Planck and Einstein, early 1900-s: Lightsometime behaves as particles, not just waves

Bohr-model, 1913

DeBroglie, 1923: Matter sometimes behave aswaves, not just particles

Schrodinger, 1932: Wave description of matter

Some ”highlights” in the evolution of quantum mechanics

Planck and Einstein, early 1900-s: Lightsometime behaves as particles, not just waves

Bohr-model, 1913

DeBroglie, 1923: Matter sometimes behave aswaves, not just particles

Schrodinger, 1932: Wave description of matter

Some ”highlights” in the evolution of quantum mechanics

Planck and Einstein, early 1900-s: Lightsometime behaves as particles, not just waves

Bohr-model, 1913

DeBroglie, 1923: Matter sometimes behave aswaves, not just particles

Schrodinger, 1932: Wave description of matter

The time independent Schrodinger Equation:

HΨ(r) = E Ψ(r)

H = − ~2

2m∇2 + V (r)

|Ψ(r)|2 gives the probability of finding the particle near position rHydrogen:Quantum numbers n, l and m, energy En = −B0

n2

n, l ,m =1, 0, 0 2, 1, 0 2, 1, 1 3, 2, 1 4, 3, 0 7, 3, 1

Light

-Consists of variouscolours/ wavelengths

Traveling electromagnetic wave

Specter

LASER

Produced through stimulatedemission

One single colour

Very intense, coherent light

• Treat the field classically

• The Dipole Approximation

E(r, t) → E(t)

The interaction between matter and light

The time dependent Schrodinger equation:

HΨ = i~∂

∂tΨ

Hamiltonian: H = Hatom + Hint

Different descriptions of the interaction:

Length gauge Velocity gauge KH frameHint = −qr · E(t) Hint = − q

mp · A(t) r → r −α(t)• Classical trajectory of a free particle in the electric field E(t):

α(t) = − q

m

∫ t

t0

∫ t′

t0

E(t ′′) dt ′′ dt ′

The Kramers Henneberger frame

HKH = − ~2

2m∇2 + V (r −α(t)) +

(q2

2mA(t)

)The particle ”sees” a moving potentialThe transition from velocity gauge to KH frame can be donewithout the dipole approximation;

HNDKH → − ~2

2m∇2 + V (r −α(r, t)) +

q2

2m(A(r, t))2

corresponding to a non-homogeneous field E(r, t)

Hyrogen atom exposed to short,intense laser pulse

E0 r

+

−

λ

Atomic Stabilisation

0 5 10 15 20 250

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Electric Field Strength (a.u.)

Ioni

zatio

n P

roba

bilit

y

High frequency

Probability of ionisation decreases with increasing intensity/frequency

Claim: Artifact of the dipole approximation

Found that the phenomena sustains inclusion of the spatialdependence of the field

The stabilistaion mechanism understood at a consequence ofthe multi-photon channel ”closing down”

Dynamics described by the time averaged KH potential:VKH(r, t) ∼ − 1

|r−α(t)| → V 0KH(r) = − 1

T

∫T

1|r−α(t)| dt

VKH(r, t) V 0KH(r)

For strong fields of high frequency, the dipole approximationbreaks downAngular distribution of the photo electron after ionization ofatomic hydrogen:

Dipole Non dipole

The interplay between the Coulomb, electric and magnetic fieldleads to the appearance of a third lobe in the angular distribution

Photo-ionisation of H+2

E

+

+R

−r

Fixed nuclei approximation

PI (R,E0) for θ = 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

1 2 3 4 5

2

4

6

8

10

R (a.u.)

E0 (a

.u.)

PI (R, θ) for E0 = 3.0 a.u.

0 1 2 3 4 5 030

60900

0.2

0.4

0.6

0.8

1

θ (degrees)Internuclear distance (a.u.)

Ioni

satio

n pr

obab

ility E

+

+R

−r

Model:

Interfering waves traveling from each of the protons:

ψout = f (Ω1)exp(ikr1)

r1+ f (Ω2)

exp(ikr2)

r2

Angular Distributions

Model: ψout → f (Ω) cos(1/2 k r · R)e ikr/r⇒ dP

dΩk∝ |f (Ω)|2 cos(1/2 k · R)

Does not give very good agreement for parallel polarisationReason: Coulomb scattering

Accounted for by the eikonal (WKB) approximation

θ = 0 θ = 45 θ = 90

k

E

R = 2a.u.

−90 −60 −30 0 30 60 900

0.01

0.02

0.03

0.04

0.05

0.06

−90 −60 −30 0 30 60 900

0.02

0.04

0.06

0.08

0.1

0.12

0.14

−90 −60 −30 0 30 60 900

0.05

0.1

0.15

0.2

R = 3a.u.

−90 −60 −30 0 30 60 900

0.05

0.1

0.15

0.2

0.25

0.3

−90 −60 −30 0 30 60 900

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

−90 −60 −30 0 30 60 900

0.05

0.1

0.15

0.2

Two photon ionisation with moving nuceli

Born-Oppenheimer approximationCorrect asymptotic behaviour in the final electronic states

Ψ(R, r, t) =∑n

∫∑νn

cn,νn(t)χνn(R)ψeln (R; r) +

∑l

∫dε

∫∑νε

c lε,νε

(t)χνε(R)ψell ,ε(R; r)

The Landau-Zener modelEnergies (adiabatic) Populations

−30 −20 −10 0 10 20 30−40

−30

−20

−10

0

10

20

30

40

Time

Ene

rgy

−30 −20 −10 0 10 20 300

0.2

0.4

0.6

0.8

1

Time (a.u.)

Pop

ulat

ion

The transition probabilities at each avoided crossing may befound analytically

The dynamcs are given by trivial phase evolution betweenavioded crossings

These ideas have been used to describe electron transport in acoupled quantum dot

Gorman et al. Phys. Rev. Lett. 95, 090502 (2005)

-Artificial molecule

4-state model:

g1 g2

e1 e2

ε

Ve

Vg

D

21

−4 −2 0 2 4

−10

−5

0

5

10

TimeE

nerg

y

E (t) = E0 sin(ωt)

In summaryStudied ionization dynamics of hydrogen• Shown that atomic stabilsation is not an artifact of thedipole approximation• Describe the phenomena accounting for dynamicstabilisation• Found signature of non-dipole interaction inphoto-ionisation in the strong field regime

Investigated the importance of geometrical factors inphoto-ionisation of H+

2 and explain its origin in terms ofinterference and refraction.• Describe angular distribution of the photoelctron includinginternuclear vibration.

Application of analytical models to describe few-level systems• Feasible method of coherent single electron transportbetween couple quantum dots

In alphabetical order

A. Palacios F. Martın I. Sundvor J. P. Hansen

J. Fernandez J. McCann L. Kocbach L. B. Madsen

M. Volkov M. Førre T. Birkeland V. Ostrovsky

The Keldysh parameter

γ =ω√

2mIp

|q|E0

0 5 10 150

0.5

1

1.5

2

2.5

3

3.5

Field strength (a.u.)

Kel

dysh

Par

amet

er

0 5 10 15 20 250

1

2

3

4

5

6

7

8

Field strength (a.u.)

Kel

dysh

Par

amet

er