ATOMIC PROCESSES IN STRONG LASER FIELDS

DEJAN B. MILOŠEVIĆ

Faculty of ScienceDepartment of PhysicsUniversity of Sarajevo

Bosnia and Herzegovina

Attosecond Science Workshop, KITP, UCSB, August 16, 2006

• Laser-assisted vs. laser-induced processes• Theory: S-matrix formalism – Feynman’s path integral –Strong-field approximation – Quantum-orbit theory• Examples• ATI by few-cycle pulses• Numerical simulations• ATI of diatomic molecules

Collaboration• B. Piraux, Ph. Antoine, A. de Bohan, UCL, LLN, Belgium (1995-1998)• Fritz Ehlotzky, University of Innsbruck, Austria (1996-)• Anthony Starace, The University of Nebraska, Lincoln, USA (1998-)• Wilhelm Becker, Max-Born-Institute, Berlin (1999-):

Alexander von Humboldt, Volkswagenstiftung• Gerhard Paulus,Texas A&M University, MPI Garching (2000-)• Dieter Bauer, MPI, Heilderberg (2004-)• Misha Ivanov (Canada) NSERC• Marc Vrakking (AMOLF, Amsterdam)• Since 2000: Research group in Sarajevo

M.Sc. Postgraduate studentsAzra Gazibegović-Busuladžić Elvedin HasovićAner Čerkić Adnan MašićSenad Odžak Aida KramoMustafa Busuladžić

• W. Becker, F. Grasbon, R. Kopold, D.B. Milošević, G.G. Paulus, and H. Walther Above-threshold ionization: from classical features to quantum effectsAdv. At. Mol. Opt. Phys. 48, 35-98 (2002)

• D.B. Milošević and F. EhlotzkyScattering and reaction processes in powerful laser fieldsAdv. At. Mol. Opt. Phys. 49, 373-532 (2003)

• D.B. Milošević, G.G. Paulus, D. Bauer, and W. BeckerAbove-threshold ionization by few-cycle pulsesJ. Phys. B 39, R203-R262 (2006)

ATOMIC PROCESSES IN A STRONG LASER FIELD

• Laser-assisted processes

- Electron-atom scattering

(Weingartshofer et al. 1977)

- X-ray-atom scattering

- Electron-ion recombination

• Laser-induced processes

- Above-threshold ionization (ATI)

(Agostini et al. 1979)

- Above-threshold detachment

- High-order harmonic generation

High-order ATI

(Paulus et al 1994)

High Harmonics

3 ,5 ,7 ,ω ω ω …14 210 W/cm , I ω>

Strong Laser Field

Ar

High-order harmonic generation (1987)

Three-step model

E

1

23

0z

nΩ ω=

( )fz t

( )iz t

KE

PI−

max 3.17P Pn I Uω = +

2/PU I ω∼

X-ray – atom scattering

DBM, F. Ehlotzky, PRA 58, 2319 (1998); Adv. AMOP 49, 373 (2003)

DBM, A. Starace, PRL 81, 5097 (1998); PRA 60, 3943 (1999); Laser Phys. 10, 278 (2000)

Electron – atom scattering

A. Čerkić, DBM, PRA 65, 053402 (2004)

Laser Phys. 15, 268 (2005)

DBM and F. Ehlotzky, PRA 65, 042504 (2002)

JMO 50, 657 (2003); Adv. AMOP 49, 373 (2003)

Electron – ion recombination

The black box of S-matrix theory

S|out> |in>

( ) ( ) ( ),

ˆ ,limfi f it t

M t U t t tψ ψ′→∞ →−∞

′ ′=

FEYNMAN’s PATH INTEGRAL

ix fx

it

ft

x

t

( )1x t

( )2x t( )clx t

i

f( )

fi

/eiS x tM ⎡ ⎤⎣ ⎦∝ ∑all paths

from i to f

( ) ( )( )f

i

, ,t

t

S dt L x t x t t= ∫

, 0S Sδ>> =Classical limit : Hamilton principle :

( ) ⇒SFA Strong Field Approximation

( ) ( )fi f, ;L3

fi f i f f fi i i/

et iS t t p

M dt d p dt H U Hψ ψ∞

−∞ −∞

∝ ∫ ∫ ∫

( ) ⇒SPM Saddle Point Method

i f

0 S S St p t

∂ ∂ ∂= = = ⇒

∂ ∂ ∂ i f, ,t t p∈

( )fi fi fiexps sM A iΦ∝ ∑relevant paths:quantum orbits s

( )

( )2

1i2

21

i i2

: HHG, HATI, HATD, NSDI: XSCA

: LAR, LAS

P

PK

II p A t

p A t

ω

⎫− ⎪

⎪ ⎡ ⎤− = +⎬ ⎣ ⎦⎪

⎡ ⎤+ ⎪⎣ ⎦ ⎭

( )I

( )II( ) ( ) ( ) ( )f

if i f i

t

tt t p A d r t r tτ τ− = ⇔ =∫

( ) ( )

( )

2 21 1

f f f2 2

21

f f 221,2

: HHG( ), LAR( ), XSCA( )

: HATI, LAS

: NSDI

P K K

j Pj

X I n

p A t p A t

p A t I

ω ω ω ′

=

⎧−⎪

⎪⎪⎡ ⎤ ⎡ ⎤+ = +⎨⎣ ⎦ ⎣ ⎦⎪

⎡ ⎤⎪ + +⎣ ⎦⎪⎩∑

( )III

S-th QUANTUM ORBIT:

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

R

i

R

f

R R i i R f

R R f f R f

Re Re

Re (HATI)

s

s

t

ns s s s st

t

s s st

r t t t p d A t t t

r t t t p d A t t

τ τ

τ τ

= − + ≤ ≤

= − + ≥

∫

∫Quantum orbits depart from

the “exit of the tunnel” real time Rt − ⇒

( ) ( )i f f0 and real r t r t t= = ≈ ⇒Orbits return almost exactly to the origin

( )RRe nsr tDBM etal, J. Mod. Opt. 53, 125 (2006)

Quantum orbits for elliptical polarization: Experiment vs. theory

ξ = 0.36

xenon at 0.77 x 1014Wcm-2

The plateau becomesa staircase

The shortest orbits arenot always the dominantorbits

Salieres, Carre, Le Deroff, Grasbon, Paulus, Walther, Kopold, Becker, Milosevic, Sanpera, Lewenstein, Science 292, 902 (2001)

• High-order atomic processes in strong fields can be explained using SFA and QO theory

• Quantum orbits are superposed in the fashion of Feynman’s path integral

• QO can be complex to account for the tun-neling nature of the initial step of the process

• Fast calculations- Usually, only very few orbits need to be considered- If solutions are found for one set of parameters then the solutions for

arbitrary parameters can be obtained by analytical continuation

• Future: application of quantum orbits to more complex strong-field processes, inclusion of the Coulomb effects, …

• Examples

( ) ( ) ( )1 2ˆ ˆc.c., / 2

2ir t is tiE t E e e E e e e x iyω ω− −

+ − ±= + + = ±

Bicircular field

( )3 1 , 1n HωΩ = ± = ±

Bicircular field

( ) ( ) ( )21 2

ˆ ˆc.c., / 22

i t i tiE t E e e E e e e x iyω ω− −+ − ±= + + = ±

Phys. Rev. A 61, 063403 (2000)

Phys. Rev. A 62, 011403(R) (2000)

Stereo-ATI experiment

Stereo-ATI experiment: Correlation technique

Stereo-ATI experiment: Experiment vs. theory

Paulus et al., Nature 2001 Milošević et al., PRL 2002

Stereo-HATI experiment: stable CE phase

Paulus et al., Phys. Rev. Lett. 91, 253004 (2003)

Milošević et al., Optics Express 11, 1418 (2003); Laser Phys. Lett. 1, 93 (2004)

HATI by bicircular ω-ω field:

Electron emission angle = CE phase / 2

Biegert, Keller

E. Hasović and DBM, Laser Phys. Lett. 3, 200 (2006)

He, 800 nm, 1015 W/cm2, CIR + Static (%)

S. Odžak and DBM, Phys. Lett. A 355, 368 (2006)

Incoherent scattering

kfATI

A A

kfHATI

ksATI

A'A

kfIS

A. Čerkić and DBM J. Phys. B submitted

Few-cycle (H)ATI

See DBM etal, J. Phys. B 39, R203 (2006)

and talk by Wilhelm Becker at conference

Vp0 = <p-eA(t)|V|0>

One cycle vs many cycles

eA(t)

p

nth cycle (n+1)st cycle (n+2)nd cycle

The discreteness of the spectrum is generated by the superposition of all cycles

The envelope is generated by the super-position of the two solutions within one cycle

energy

Examples of direct quantum orbits

One member of a pair of orbits experiences the Coulomb potential more than the other

Interference of the two solutions from within one cycle(includes focal averaging)

F-

λ = 1500 nm

cf. M.V. Frolov, N.M. Manakov, E.A. Pronin,A.F. Starace,JPB 36, L419 (2003)

Data: I. Yu Kiyan, H. Helm, PRL 90, 183001 (2003) (1.1 x 1013 Wcm-2 )Theory: D.B. Milosevic et al., PRA 68, 070502(R) (2003) (1.3 x 1013 Wcm-2 )

Rescattered quantum orbits in space and time

ionization time = t´ t = recollision time

Few-cycle-pulse ATI spectrum: violation of backward-forward symmetry

Different cutoffs

Peaks vs no peaks

argon, 800 nm7-cycle durationsine square envelopecosine pulse, CEP = 01014 Wcm-2

D. B. Milosevic, G. G. Paulus, WB, PRA 71, 061404 (2005)

Building up thespectrum fromquantum orbits

The black box of S-matrix theory ...

|out> = S|in>

S|p>

|0>

... has been made transparent

|p>|0>

ATI simulation

• S-matrix theory• SFA (1BA) with rescattering (2BA)• Physical interpretation via Quantum Orbits• Focal averaging + saturation effects• Gauge aspects are important (see WB)• Choice of the ground-state wave function• Coulomb effects for ATI (not included yet)

QM transition amplitude

( ) ( ) ( ) ( )0

lim ,t

pi p itM i dt t U t t r E t tψ ψ

→∞′ ′ ′ ′= − ⋅∫Exact:

( ) ( )0 1pi pi piM M M= + +SFA:

( ) ( ) ( ) ( ) ( )0 00 0 0 0

0

exppT

pi i pM i dt p A t r E t iS tψ ⎡ ⎤= − + ⋅ ⎣ ⎦∫

( ) ( ) ( )

( ) ( ) ( ) ( )0

1 30 1 1 1

0

10 0 0 1 exp , ,

pT

pit

i p

M dt dt d k p A t V k A t

k A t r E t iS t t kψ

∞

= − + +

⎡ ⎤× + ⋅ ⎣ ⎦

∫ ∫ ∫

Hartree-Fock ground-state wave functions, expanded on basis set of atomic Slater orbitals:

( ) ( )10

ˆj jn ri j j l

j

r C N r e Y rςψ −= ∑

Independent-particle-model optimized potential, represented by the double Yukawa potential:

( ) ( )/

/ 0.41 1 , r D

Hr DZ eV r H e H DZH r

−−⎡ ⎤= − + − =⎣ ⎦

He (1s, Z=2), Ne, Ar, Kr, Xe (5p, Z=54)

He, 1015 W/cm2, 760 nm

Xe, 8.05x1013 W/cm2, 760 nm

( )1 21 lpiT M M∝ + −

SPM for HATI and QO theory

Classification of saddle-point solutions: αβm

The uniform approximation

Semiclassical analysis. Cutoff lawBusuladzic et al, Laser Phys. , 289 (2006)16 max 10.007 0.538p p pE U I= +

• Enhancement of HATI at channel closings

Gaussian focal averagingKopold et al, J. Phys. B 35, 217 (2002)

( ) ( )max 1/ 2

max

0

lnI

pi pi p p pn

IdIw w n E I U nI I

δ ω⎛ ⎞∝ + + −⎜ ⎟⎝ ⎠

∑∫

Improved algorithm which includes saturation effects – important for new Br- ATD experiments

( )

( )( ) ( )( )

20

,4

; , exp ' , '

plm p pn

t

plm lm

I tw d dt E I n

w n I t dt w I t

ρρρ δ ω

ω

ρ ρ

∞ ∞

−∞

−∞

⎛ ⎞∝ + + −⎜ ⎟⎜ ⎟

⎝ ⎠⎛ ⎞

× −⎜ ⎟⎝ ⎠

∑∫ ∫

∫

No averaging Focal averaging

24p pIE n Iωω

= − − ( )p cE n n ω= −c p pn I Uω = +nc-photon channel is closed

• Resonant-like enhancement at particular I• Channel-closing effect in HATI• Near channel closing Ep is low• Many QO interfere constructively• All intensities from I=0 to I=Imax contribute to the focal-averaged electron yield• Channel-closing intensities are:

( ) 2

,min ,min ,max

,min ,max max

4

, 1, ,

/ 1,

c c p

c c c c

c p c

I n I

n n n n

n I I I

ω ω

ω

= −

= +

⎡ ⎤= + ≤⎣ ⎦

…

enhancement regions – a series of rounded tops in the envelope of the electron spectra

,max ,min 1c cn n− +

Intensity-dependent enhancements

intensityincreasesby 6%

Hertlein, Bucksbaum, Muller, JPB 30, L197 (1997) Paulus et al., PRA 64, 021401 (2001)

see, also, Hansch, Walker, van Woerkom, PRA 55, R2535 (1997)

Physical origin of the enhancements

Constructive interference of long quantum orbits

Quantum effect!!!

For zero drift momentum,p = 0, the electron revisitsinfinitely often

p2/(2m) = Nω − Ip – Up = 0

Ip + Up = Nω„Channel Closing“

Analytical proof:S.V. Popruzhenko, P.A. Korneev, S.P. Goreslavskii, WB, PRL 89, 023001 (2002)D.B. Milosevic, WB, PRA 66, 063417 (2002)

Alternative explanations of the intensity-dependent enhancements

Threshold cusps a la Wigner/Baz:

B. Borca, M.V. Frolov, N.L. Manakov, A.F. Starace, PRL 88, 193001 (2002)

Multiphoton resonance with ponderomotively upshifted Rydberg state:

H.G. Muller, F.C. Kooiman, PRL 81, 1207 (1998)H.G. Muller, PRL 83, 3158 (1999)J. Wassaf, V. Veniard, R. Taieb, A. Maquet, PRL 90, 013003 (2003)

Enhancements disappear for short pulses

pulse duration

enhancements disappearfor short pulses

peaks become narrower for long pulses

Focal-averaged angular-resolved ATI energy spectrum xenon, 760 nm, 8 x 1013 Wcm-2

contains the effects of

channel closings(several)

cutoff (rainbow)enhancements

con-/destructive interferences ofpairs of quantum orbitsand/or the twomembers of each pair

Even (odd) CC vs. s (p) ground states

Manakov and Frolov, JETP Lett. 83, 536 (2006)

Krajewska, Fabrikant, and Starace, Phys. Rev. A (submitted)

DBM etal (in preparation)

Conclusions• Realistic choices of the initial wave function and

the rescattering potential (improved SFA)• Plateau height: direct vs. rescattering

He (6 orders), Xe, Kr (1-2 orders)• Plateau length:

Direct: s-state (2-3Up), p-state (4Up => interf.) Rescattering:

• Quantum orbits αβm, θ – dependent cutoff• Focal averaging and saturation• Qualitative change due to channel closing eff.• CC – pronounced enhancement• Agreement with experimental results

max 10.007 0.538p p pE U I= +

Ionization of a diatomic molecule by a strong laser field

( ) ( ) ( )22 2

tot , ,2 2 2

eA Be A A B B e A B

A B e

pP PH er e R e R E t V r R RM M m

= + + − + + ⋅ +

( ) ( )Jacobi

, , , , , , , ,e A B CM tr R R r R R i r R t H r R tΦ Φ→ ∂ =→

( ) ( ) ( )2 2

,2 2 r RP pH e r e R E t V r R

mµ= + − + ⋅ +

( ) ( ) ( ) ( ), A Be A e B ABV r R V r V r V R= + +

( ) ( )CM CM lim lim ,fi f i fit tS P P M t tδ

′→∞ →−∞′= −

( ) ( ) ( )

( ) ( ) ( )

3 3 * 3 3, , , , , ,

, ,

t

fi ft

r R i

M t t i d d r d R r R t d r d R r R U t r R

e r e R r R

τ Φ τ

Ε τ Φ τ′

′ ′ ′ ′ ′=

′ ′ ′ ′× + ⋅

∫ ∫ ∫ ∫ ∫

Neglect e-a int: F FF e ABH H h H→ = +

( ) ( ) ( )2 2

, 2 2

F Fe r AB R AB

p Ph e r E t H e R E t V Rm µ

= − ⋅ = − ⋅ +

( ) ( ) ( )* * * *SFA : , , , ABv f

f f

iE tf f ep ABvU r R t r t R eΦ Φ φ ϕ→ =

( ) ( ) ( ) ( )( )BOA : , , ; ei ABvi

i

i E R E ti ei ABvr R t r R R eΦ φ ϕ +

=

( ) ( ) ( )

( ) ( ) ( )( )

( )

0

,

1/ 20 1

MO-LCAO : ;

2HFR-STO: ,

2 !

a

a a

a a

ei Ja a JJ A B a

na n r

a J l m

a

r R c r

r r e Yn

ς

φ ψ

ςψ θ ϕ

=

+

− −

=

=

∑ ∑

( ) ( )( ) ( )CM CM 202 / 2

f i f ffi f i v v f ABv ABv ei p fin

S i P P S p E E E R U n T nπ δ δ ω= − − + − − + −∑

g 23 HOMO of Nσ

g 21 HOMO of Oπ

N2

O2

g 21 HOMO of Fπ g 23 HOMO of Fσ