Phase and amplitude in molecular high-harmonicgeneration

Manfred Lein

Collaborators: Elmar van der Zwan, Ciprian Chirila,

at University of Wurzburg: Mirjam Falge, Volker Engel

– p.1

Outline

• Two-center interference and core dynamics in H2

• High-harmonic generation in H2O

• Phase jump in the orientation dependence of theharmonic phase

• Wave-packet study of two-center interference

– p.2

High-harmonic generation

HHG process

ωωωωω

Ω = Νω

ω

N photons of frequency ω

→ 1 photon of frequency Nω.

– p.3

High-harmonic generation

HHG process

ωωωωω

Ω = Νω

ω

N photons of frequency ω

→ 1 photon of frequency Nω.

Recollision picture

V(x)

V(x)

V(x)

x x x

Ionization Free acceleration Recombination

Maximum return energy:

Emax = 3.17Up

→ Cut-off at ~ω = 3.17Up + Ip

[Corkum, Schafer,. . . 1993]

Harmonic order

plateau cut−off

– p.3

Two-center interference in molecules

Recolliding electron with wave vector k

TDSE for H2 in 2D

with vibrationkk

λλ/2

destructiveconstructiveinterference interference

0 1 2 3 4Photon energy (a.u.)

10−8

10−6

10−4

10−2

Inte

nsity

(ar

b. u

.) θ=0ο

M.L., N. Hay, R. Velotta, J.P. Marangos, P.L. Knight, PRL 88, 183903 (2002),

Experimental confirmation for CO 2:T. Kanai et al., Nature 435, 470 (2005)C. Vozzi et al., PRL 95, 153902 (2005)

– p.4

Conditions for two-center interference

• Plane wave for continuum state

• LCAO for bound state

• velocity form matrix element

→ Recombination described by⟨

exp(ik·r)∣

∣k∣

∣Ψ⟩

=⟨

exp(ik·r)∣

∣k∣

∣φ(r− R/2)+φ(r + R/2)⟩

= 2 cos(k · R/2)⟨

exp(ik·r)∣

∣k∣

∣φ⟩

– p.5

Influence of molecular vibration in HHG

Illustration of physical mechanism:

0 1 2 3 4 5R (a.u.)

χ0(R)χ(R,τ)

H2

+

H2

Harmonics for electron travel time τ are approximately proportional to

|autocorrelation function|2 = |C(τ)|2 = |∫

χ∗(R, 0)χ(R, τ)dR|2

→ In general, vibration reduces efficiency of HHG.

M.L., PRL 94, 053004 (2005).

– p.6

Influence of molecular vibration in HHG

Putting it together: S(ω) ∼

∣

∣

∣

∣

∫

χ∗(R, 0) cos(k · R/2)χ(R, τ)dR

∣

∣

∣

∣

2

– p.7

Influence of molecular vibration in HHG

Putting it together: S(ω) ∼

∣

∣

∣

∣

∫

χ∗(R, 0) cos(k · R/2)χ(R, τ)dR

∣

∣

∣

∣

2

Experiment:

S. Baker et al., PRL 101, 053901 (2008)

I = 3 × 1014 W/cm2

I = 2.2 × 1014 W/cm2

– p.7

HHG in water molecules

Attractive features of H 2O

• electron in HOMO and nuclear motion nearly decoupled

• rapid vibrational motion

HH

O

HOMO

Short time scales:→ only vibrational degrees relevant

– p.8

Vibrational states of H2O

Nuclear wavefunctions of H2O

1 1.5 2 2.5 3 1

1.5

2

2.5

3

0 0.5 1 1.5 2 2.5 3 3.5

groundstate, E=0.4604 eV

1 1.5 2 2.5 3 1

1.5

2

2.5

3

0 0.5 1 1.5 2 2.5 3

n=1, E=0.9055 eV

1 1.5 2 2.5 3 1

1.5

2

2.5

3

0 0.5 1 1.5 2 2.5 3

n=2, E=0.9084 eV

1 1.5 2 2.5 3 1

1.5

2

2.5

3

0 0.5 1 1.5 2 2.5 3

n=3, E=1.3338 eV

1 1.5 2 2.5 3 1

1.5

2

2.5

3

0 0.5 1 1.5 2 2.5

n=4, E=1.3343 eV

1 1.5 2 2.5 3 1

1.5

2

2.5

3

0 0.5 1 1.5 2 2.5

n=5, E=1.3540 eV

– p.9

Autocorrelation functions

Comparison of the correlations of H2O and D2O

0

0.2

0.4

0.6

0.8

1

0 1 2 3 4 5

|cor

rela

tion|

t/fs

groundstate wavefunction

H2OD2O

0

0.2

0.4

0.6

0.8

1

0 1 2 3 4 5

|cor

rela

tion|

t/fs

1st excited state wavefunction

H2OD2O

0

0.2

0.4

0.6

0.8

1

0 1 2 3 4 5

|cor

rela

tion|

t/fs

2nd excited state wavefunction

H2OD2O

0

0.2

0.4

0.6

0.8

1

0 1 2 3 4 5

|cor

rela

tion|

t/fs

3rd excited state wavefunction

H2OD2O

– p.10

Autocorrelation functions

Comparison of the correlations of H2O and D2O

0

0.2

0.4

0.6

0.8

1

0 1 2 3 4 5

|cor

rela

tion|

t/fs

4th excited state wavefunction

H2OD2O

0

0.2

0.4

0.6

0.8

1

0 1 2 3 4 5

|cor

rela

tion|

t/fs

5th excited state wavefunction

H2OD2O

0

0.2

0.4

0.6

0.8

1

0 1 2 3 4 5

|cor

rela

tion|

t/fs

6th excited state wavefunction

H2OD2O

0

0.2

0.4

0.6

0.8

1

0 1 2 3 4 5

|cor

rela

tion|

t/fs

7th excited state wavefunction

H2OD2O

– p.11

Ratios of autocorrelations

Ratios of the absolute value of the correlation of D2O and H2O

0.4 0.6 0.8

1 1.2 1.4 1.6 1.8

2 2.2

0 1 2 3 4 5

C(D

2O)/

C(H

2O)

t/fs

groundstate wavefunction

0.4 0.6 0.8

1 1.2 1.4 1.6 1.8

2 2.2

0 1 2 3 4 5

C(D

2O)/

C(H

2O)

t/fs

1st excited state

0.4 0.6 0.8

1 1.2 1.4 1.6 1.8

2 2.2

0 1 2 3 4 5

C(D

2O)/

C(H

2O)

t/fs

2nd excited state

0.4 0.6 0.8

1 1.2 1.4 1.6 1.8

2 2.2

0 1 2 3 4 5

C(D

2O)/

C(H

2O)

t/fs

3rd excited state

0.4 0.6 0.8

1 1.2 1.4 1.6 1.8

2 2.2

0 1 2 3 4 5

C(D

2O)/

C(H

2O)

t/fs

4th excited state

0.4 0.6 0.8

1 1.2 1.4 1.6 1.8

2 2.2

0 1 2 3 4 5

C(D

2O)/

C(H

2O)

t/fs

5th excited state

– p.12

Questions

• Do these models hold in calculations for multielectron system?

• What is the effect of the laser field on the core?

→ Models:

• Strong-field approximation (SFA) including vibration and dressing ofthe ion

• Exact solution of the time-dependent Schrödinger equation (TDSE)for a 1D H2 model system with 2 interacting electrons

– p.13

Harmonic generation in H2

Influence of core dynamics on HHG: vibration + dressing

0 1 2 3 4 5R (a.u.)

χ0(R)χ(R,τ)

H2

+

H2

σu

σg

– p.14

SFA with laser dressing of the ion

Wave-packet dynamics in the two coupled lowest BO states

i ∂∂t

χu(R)

χg(R)

=

Tn + Vu(R) E(t) · D(R)

E(t) · D(R) Tn + Vg(R)

χu(R)

χg(R)

incorporated in the Lewenstein model for vibrating H2:

P(t) = −2it∫

0

dt′ exp(−iS(ps, t, t′))

[

2πǫ+i(t−t′)

]3/2

×

∫

dR χ0(R)

prec,u(R)

prec,g(R)

T

Uion(t, t′)

dion,u(R)

dion,g(R)

χ0(R) + c.c.

– p.15

Results 800 nm

H2/D2 in 800 nm 14-cycle pulses (parallel alignment), I = 5 × 1014 W/cm2

no dressing dressing

0 20 40 60 80Harmonic order

0

1

2

3

ratio

D2/

H2

0 20 40 60 8010

−10

10−5

100

|ax(

Ω)|

2

0 20 40 60 80Harmonic order

0

1

2

30 20 40 60 8010

−10

10−5

100

C.C. Chirila, M.L.,PRA 77, 043403 (2008)

Ratios D2/H2: full/ short /long trajectories

– p.16

Results 2000 nm

H2/D2 in 2 µm 14-cycle pulses (parallel alignment), I = 5 × 1014 W/cm2

no dressing dressing

0 500 1000Harmonic order

0

1

2

3

ratio

D2/

H2

0 500 100010

−12

10−10

10−8

10−6

10−4

|ax(

Ω)|

2

0 500 1000Harmonic order

0

1

2

30 500 100010

−12

10−10

10−8

10−6

10−4

C.C. Chirila, M.L.,PRA 77, 043403 (2008)

Ratios D2/H2: full/ short /long trajectories– p.17

One-dimensional model

Solve TDSE i∂Ψ∂t = HΨ for

(i) “2e-calculation”: H = Tn + Te +1

R+ Wen(R, x1) + Wen(R, x2)

+Wee(x1 − x2) + (x1 + x2)E(t)

with

Tn = −1

M

∂2

∂R2, Te = −

1

2µe

(

∂2

∂x21

+∂2

∂x21

)

Wen(R, xj) = −∑

s=±1

1√

(xj + sR2 )2 + β2(R)

, Wee(x) =1

√

x2 + α2(R).

α(R),β(R) adjusted to reproduce BO potentials of H2 and H+2

[Sagout et al., PRA 77, 023404 (2008)]

– p.18

One-dimensional model

Solve TDSE i∂Ψ∂t = HΨ for

(i) “2e-calculation”: H = Tn + Te +1

R+ Wen(R, x1) + Wen(R, x2)

+Wee(x1 − x2) + (x1 + x2)E(t)

with

Tn = −1

M

∂2

∂R2, Te = −

1

2µe

(

∂2

∂x21

+∂2

∂x21

)

Wen(R, xj) = −∑

s=±1

1√

(xj + sR2 )2 + β2(R)

, Wee(x) =1

√

x2 + α2(R).

α(R),β(R) adjusted to reproduce BO potentials of H2 and H+2

[Sagout et al., PRA 77, 023404 (2008)]

(ii) “1e calculation”: H ′ = H − x2E(t)

– p.18

Results 1500nm

0 50 100 150 200Harmonic order

10−10

10−8

10−6

10−4

10−2

100

Inte

nsity

H2, 1500nm, 2x1014

W/cm2, 2−3−2 pulse

two−center formula

Shift of interference minimum consistent with nuclear motion

– p.19

Results 800nm

Ratios D2/H2, 2-3-2 pulses

0 10 20 30 40Harmonic order

0.5

1.0

1.5

2.0

2.5

Rat

io D

2/H

2

0.5

1.0

1.5

2.0

2.5

Rat

io D

2/H

2

0 10 20 30 40 50Harmonic order

0.5

1.0

1.5

2.0

2.5

0.5

1.0

1.5

2.0

2.5a) 1.5x1014

W/cm2

b) 2.0x1014

W/cm2

c) 2.5x1014

W/cm2

d) 3.0x1014

W/cm2

2e-calculation

1e-calculation

Ratios of˛

˛

˛

˛

R

χ∗(R, 0) cos(k · R/2)χ(R, τ)dR

˛

˛

˛

˛

2

→ Dressing not important for 800 nm

– p.20

Phase jump in molecular HHG

Amplitude/phase dependence on molecular orientation for 2D molecules:

0 30 60 90Angle (degrees)

−π/2

0

π/2

Pha

se

31 79

79

7171

6110

−1

100

Am

plitu

de

H2

+

0 30 60 90Angle (degrees)

π/2

−π/2

0

10−1

100

H2

polarization parallel to laser field

polarization perpendicular to laser field

M.L., N. Hay, R. Velotta,J.P. Marangos, P.L. Knight,PRL 88, 183903 (2002).

Destructive two-center interference goes along with a phase jump.

Size of the jump: ∆ϕ<∼ π.

– p.21

Phase jump in molecular HHG

Measurement of emission time/phase of harmonics from CO2:

Boutu et al., Nature Physics 4, 545 (2008)

Phase jump forCO2|| laser field:• less than π• smeared out

See also:Wagner et al., PRA 76,061403 (2007)

Zhou et al., PRL 100,073902 (2008)

– p.22

SFA revisited

Different versions of SFA (all using plane waves)

• Full SFA:

P(t) =t∫

0

dt′E(t′)∫

d3p 〈p+A(t′)|x|Ψ〉 〈Ψ|∇|p+A(t)〉 e−iS(p,t,t′) + c.c.

includes trajectories not parallel to driving field.

• SFA in saddle-point approximation for momentum (SP):only trajectories parallel to field,all ionization times contribute to one harmonic.(“conventional SFA”)

• SFA in saddle point approximation for momentum + time:harmonic order determines ionization/recombination time andrecollision momentum.

– p.23

SFA revisited

Problems of conventional SFA

1) No harmonics when field || nodal plane in symmetrical orbital

+

E

−

〈Ψ|Ex|eikx〉 = 0.

2) In velocity form no harmonics polarized ⊥ laser field (for any orbital)

〈Ψ|∂y|eikx〉 = 0

– p.24

Effect of non-saddle point dynamics

Comparison of full and saddle-point SFA

0 30 60 90Angle (degrees)

0

π/2

π

Pha

se

10−13

10−12

10−11

10−10

10−9

10−8

Inte

nsity

780 nm, 5 x 1014

W/cm2, harmonic 43

exactSP

Phase jump is• smooth in both calculations• smoother in the full SFA (including nonparallel trajectories).

– p.25

Effect of non-saddle-point dynamics

Orientation dependence for H+2 ungerade 1st excited state

(adjusted to same Ip)

0 30 6010

−13

10−11

10−9

Am

plitu

de

π

3π/2

2π

5π/2P

hase

order 41

exactSP

0 30 60Angle degrees

order 61

0 30 60 90

order 83

– p.26

Effect of non-saddle-point dynamics

Orientation dependence for H+2 ungerade 1st excited state

(adjusted to same Ip)

0 30 6010

−13

10−11

10−9

Am

plitu

de

π

3π/2

2π

5π/2P

hase

order 41

exactSP

0 30 60Angle degrees

order 61

0 30 60 90

order 83

→ Saddle-point SFA fails near 90o.– p.27

Wave-packet study of two-center interference

Motivation:

Speculation about shift of the Cooper minimum in HHG from Argon withrespect to photoionization cross sections (M. Gühr et al.)

→ Compare interference minima

• in laser-driven HHG

• in laser-field-free recollision of a Gaussian wave packet

Model: 2D H+2 with fixed nuclei

– p.28

Wave-packet study of two-center interference

0 20 40 60 80Harmonic order

0

30

60

90A

ngle

at m

inim

um

780 nm, 5x1014

W/cm2

Gaussian wave packet2−center model2−center model (Ip corrected)laser pulse

– p.29

Conclusions

• Ratio D2/H2 at 800 nm well explained by

two-center interference +nuclear motion.

• Laser dressing in H2 important only for long-wavelength fields.

• Isotope effects in H2O vs D2O increase for HHG from

vibrationally excited initial state

• Orientation dependence of phase:

phase jumps smeared out by non-saddle-point dynamics,

• Wave packet study indicates negligible laser-induced shift ofinterference minimum

– p.30