Quantum Optics VII, June 8 – 12, 2009, Zakopane, Poland

The Zero-Range Potential in an IntenseLaser Field: Much more than a Toy

W. Becker1 and D. B. Milosevic1,2

1Max-Born-Institut, Berlin, Germany2Faculty of Science, University of Sarajevo, Sarajevo,

Bosnia and Herzegovina

c2/2 ac

An effective-source model for above-threshold ionization

Volkov solution

regularized ZRP in an arbitrary number of spatial dimensions

in any number of dimensions but one, the ZRP (x) mustbe regularized to allow for a normalizable bound state:

Zero-range potentials play an important role in quantum physics because they provide the simplest (and sometimes even explicitly solvable) models of complicated interactions.But

(x) reg(x)

Alternatively, a separable potential V = |> <| has a single boundstate for appropriate

B.G. Englert, Lett. Math. Phys. 34, 239 (1995)K. Krajewska, J.Z. Kaminski, K. Wodkiewicz, Opt. Commun., to be published

regularized ZRP in an arbitrary number of spatial dimensions

E0 = - ma2/2

d = 5

d = 2 E0 = -2 exp[-2/a2

+ (1)]/L2

Many applications in one and three dimensions

neutron-proton scattering (Fermi 1936)nuclear physicssolid state Kronig-PenneyBE condensates with attractive short-range interactionmodeling the tip in scanning-tunneling microscopyelectrons bound in negative ionslaser-atom physics

large wave length (s-wave) scatteringvery simplest model atom (one bound state plus continuum)

Applications in two dimensions

2d Kronig-Penney

chaotic quantum billiard (P. Seba 1990)

2d electron gas: Imaging magnetic focusing of coherent electron waves (K.E. Aidala, Nature Physics 3, 484 (2007))

Applications of ZRPs to intense-laser-atom physics

N.L. Manakov and L.P. Rapoport, Sov. Phys. JETP 42, 430 (1976)I.J. Berson, J. Phys. B 8, 3078 (1975)N.L. Manakov and A.G. Fainshtein, Sov. Phys. JETP 52, 382 (1981)W. Elberfeld and M. Kleber, Z. Phys. B 73, 23 (1988)W. Becker, S. Long, and J.K. McIver, Phys. Rev. A 41, 4112 (1990)F.H.M. Faisal, P. Filipowicz, and K. Rzazewski, Phys. Rev. A 41, 6176 (1990)P. Filipowicz, F.H.M. Faisal, and K. Rzazewski, Phys. Rev. A 44, 2210 (1991)P.S. Krstic, D.B. Milosevic, and R.K. Janev, Phys. Rev. A 44, 3089 (1991)W. Becker, J.K. McIver, and K. Wodkiewicz, Laser Phys. 3, 475 (1993)J.Z. Kaminski, Phys. Rev. A 52, 4976 (1995)

B. Borca, M.V. Frolov, N.L. Manakov, and A.F. Starace, PRL 87, 133001 (2001)

Evolution of the atomic ground state in a laser field

)''()'()'';(''),( 3 tVttGddtt Volkovt

rrrrrr

The wave function within the range of V(r) determines the wave function everywhere

)'||exp()'()''( 00 tEit rr

„direct ionization“ (no modification of the ground-statewave function by the laser field)

(virtually exact for a circularly polarized field)

Evolution of the atomic ground state in a laser field

)''()'()'';(''),( 3 tVttGddtt Volkovt

rrrrrr

The wave function within the range of V(r) determines the wave function everywhere

)'||exp()'()''( 00 tEit rr

„direct ionization“ (no modification of the ground-statewave function by the laser field)

(virtually exact for a circularly polarized field)

propagation in the laser field only, no potential

Evolution of the atomic ground state in a laser field

)''()'()'';(''),( 3 tVttGddtt Volkovt

rrrrrr

The wave function within the range of V(r) determines the wave function everywhere

Insert the integral equation into itself:

)''||exp()''()''()'''',''(''''

)'()'';(''),(

003

'

3

tEiVttGddt

VttGddtt

Volkovt

Volkovt

rrrrr

rrrrr

allows for a modification of the wave function within the rangeof V(r) due to the laser field (max. one act of rescattering)

Overview of the rest of the talk:

The ZRP in a laser field, as it is, describes many experiments even semiquantitatively and generates surprisingly complex spectra

Overview of the rest of the talk:

The ZRP in a laser field, as it is, describes many experiments even semiquantitatively and generates surprisingly complex spectra

For atom-specific results, atom-specific (non ZRP) potentials must be introduced (adjusting the ionization energy is not enough)

Overview of the rest of the talk:

The ZRP in a laser field, as it is, describes many experiments even semiquantitatively and generates surprisingly complex spectra

For atom-specific results, atom-specific (non ZRP) potentials must be introduced (adjusting the ionization energy is not enough)

For ionization off the laser-polarization direction, the lowest-order Born approximation becomes insufficient

ZRP high-order harmonic spectra

various rare gases, I = 3 x 1013 Wcm-2, = 1.16 eV

WB, S. Long, J. K. McIver, PRA (1990)

Intensity-dependent enhancements of groups of above-threshold-ionization peaks in the

rescattering regime

Intensity-dependent enhancements

Hertlein, Bucksbaum, Muller, JPB 30, L197 (1997) Paulus, Grasbon, Walther, Kopold, Becker, PRA 64, 021401 (2001)

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

intensityincreasesby 6%

0.5 I0

1.0 I0

S-matrix element for ionization from the ground state |0> into a continuum state |p> with momentum p

)(|),(|)( 0

)()(iif

Volkovf

Volkovt

if tVttVUtdtdtMi

pp

|p(t) > = Volkov state, U(Volkov)(t,t‘) = Volkov propagator

V = binding potential

)'()()',( 3)( ttdttU Volkovkkk

Saddle-point (steepest-descent) evaluation of the amplitude

f

i

ff

t

tip

t

ifp

t

if

tIedi

edi

ttkmddtdtM

2

23

))((2

exp

))((2

exp),,(

Ak

Apkp

Find values of k, tf, and ti, so that the exponentials be stationary:

0.../.../.../ if ttk

saddle-points equs. with infinitely many (complex) solutions ks, tfs, tis (s=1,2,...)

pi Ite 2))(( 2 Ak 22 ))(())(( ff tete ApAk

f

i

ttif edtt )()( Ak

M. Lewenstein, Ph. Balcou, M.Yu. Ivanov, A. L‘Huillier, and P.B. Corkum, Phys. Rev. A 49, 2117 (1994)M. Lewenstein, K.C. Kulander, K.J. Schafer, and P.H. Bucksbaum, Phys. Rev. A 51, 1495 (1995)

pi Ite 2))(( 2 Ak

22 ))(())(( ff tete ApAk

f

i

ttif edtt )()( Ak

Saddle-point equations

elastic rescattering

f

i

tt i fe d t t) ( ) ( A k

return to the ion

tunneling at constant energy

Quantum-orbit expansion of the ionization amplitude

),,(exp),,( isfssisfsss

ttiSttmM kk ppp

coherent superposition of different pathways into the same final state

+ + + +....

realization of Feynman‘s path integraleach orbit by itself depends only very smoothly on intensity

Salieres et al., Science 292, 902 (2001)

NB: coherent-superposition effects are quantum effects

Enhancements: SFA-type theory vs experiment

argon spectra, 6.45 1013 Wcm-2< I < 6.88 1013Wcm-2

Hertlein, Bucksbaum, and Muller, JPB 30, L197 (1997)

6.39 1013Wcm-2< I < 6.91 1013Wcm-2

Kopold, Becker, Kleber, Paulus, JPB 38, 217 (2002)

Focal-averaged zero-range potential SFA simulation

Physical origin of the enhancement

Electron energies: Ep = p2/(2m) = (N+n) - Ip - Up

At the channel-closing intensity: Up + Ip = N,

Electrons are emitted with zero drift momentum, p = 0 (n=0).

Constructive interferenceof long quantum orbits

Many recurrences: many opportunities for rescattering

Quantum effect!!!

(At channel closings: Ep = n)

ATI channel-closing (CC) enhancements

electron energy = 199 eV, Ti:Sa laser, He, 1.04 x 1015 Wcm-2 < I < 1.16 x 1015 Wcm-2

number of quantumorbits included in the calculation

a few orbits aresufficient toreproduce thespectrum,except near CCs

D.B. Milosevic, E. Hasovic, M. Busuladzic,A- Gazibegovic-Busuladzic, WB, Phys. Rev. A 76, 053410 (2007)

ATI channel-closing (CC) enhancements

electron energy = 199 eV, Ti:Sa laser, He, 1.04 x 1015 Wcm-2 < I < 1.16 x 1015 Wcm-2

number of quantumorbits included in the calculation

a few orbits aresufficient toreproduce thespectrum,except near CCs

Constructive interference of many long orbits

This explains:

Resonantlike enhancements occur at channel closings,preferably (for even-parity ground state) with even-integer order (Wigner‘s threshold law)

Enhancements are restricted to approx. 4Up < Ep < 8Up

No enhancements for „direct“ electrons

Enhancements occur for one or several groups of ATI peaks, each comprising about 8 peaks (Ti:Sa)

Magnitude of the enhancements decreases with increasing intensity

Enhancements vanish for short pulses

Alternative explanations

Solution of the 3D TDSE: H. G. Muller and F. C. Kooiman, PRL 81, 1207 (1998); H. G. Muller, PRA 60, 1341 (1999); PRL 83, 3158 (1999)

Wigner-Baz threshold effect for 3D zero-range (short-range) potential;B. Borca, M. V. Frolov, N. L. Manakov, A. F. Starace, PRL 88, 193001 (2002)

Solution of the 1D TDSE vs. multiphoton resonance with Floquet quasienergy states vs. trajectories with nonzero initial velocity: J. Wassaf, V. Veniard, R. Taieb, A. Maquet, PRL 90, 013003 (2003); PRA 67, 053405 (2003)

3D TDSE vs. Floquet quasienergies: R. M. Potvliege and S. Vucic, PRA 74, 023412 (2006)

3D R-matrix Floquet:K. Krajewska, I. I. Fabrikant, A. F. Starace, PRA 74, 053407 (2006)

Quantitative rescattering theory:

Reconstruction of the electron-ion potential

recolliding electron has momentum p = (2 x 3.17 Up)1/2 and rescatters elastically according to the electron-ion cross section

Is ATI good for something?

recolliding electron has momentum p = (2 x 3.17 Up)1/2 and rescatters elastically according to the electron-ion cross section

pfx = -- A(tr) + p cos r pfT = p sin r

Is ATI good for something?

recolliding electron has momentum p = (2 x 3.17 Up)1/2 and rescatters elastically according to the electron-ion cross section

picks up the additional momentum - A(tr) from the field after rescattering

pfx = -- A(tr) + p cos r pfT = p sin r

M. Okunishi, T.Morishita, G. Pruemper, K. Shimada, C. D. Lin, S. Watanabe, K. Ueda, PRL 100, 143001 (2008)

Is ATI good for something?

D. Ray, B. Ulrich, I. Bocharova, C. Maharjan, P. Ranitovic, B. Gramkov, M. Magrakvelidze, S. De., I.V. Litvinyuk,A.T. Le, T. Morishita, C.D. Lin, G.G. Paulus, C.L. Cocke, PRL100, 143002 (2008)

the same for xenon

theory: effective model potential, Coulomb + short range

M. Okunishi et al., PRL 100, 143001 (2008)

Formal description of recollision processes

|)'()(|)',(

)'(|)'()',(|)('

)(|)(|)(

3

0

0

)1()0(

ttdttU

ttttUVtdtdt

tttdti

MMM

VolkovVolkovVolkov

Volkovff

t

f

fififi

qqq

Er

Er

= „direct“ + rescattered

Formal description of recollision processes

|)'()(|)',(

)'(|)'()',(|)('

)(|)(|)(

3

0

0

)1()0(

ttdttU

ttttUTtdtdt

tttdti

MMM

VolkovVolkovVolkov

Volkovff

t

f

fififi

qqq

Er

Er

going beyond the first-order Born approximation

ff TV fVffff VGVVT

A. Cerkic, E. Hasovic, D.B. Milosevic, WB, PRA, 79, 033413 (2009)

Low-frequencyapproximationLFA

First-order Born vs. Low-Frequency Approximation

LFA generates zeros in the differential cross section

Ar

2.3x1014 Wcm-2

800nm

crosssection

Comparing the calculated electron-argon+ cross sectionwith the cross section extracted from HATI calculations

calculated extracted from HATI

Comparison of first-order Born vs Low-Frequency approximation

1BA

LFA

High-order above-thresholdionization

xenon at 1.5x1014 Wcm-2 760 nm

in the momentum (px,pz) plane

laser pol.direction

1BA is only sufficient(if at all) in the directionof the laser polarization

Conclusion

The zero-range potential provides a perfect model forthe laser-atom interaction

Improvements allow for an atom-specific quantitativedescription of ionization spectra

Thank you, Krzysztof, for very many yearsof friendship and inspiration