Quantum-orbit approach for an elliptically polarized laser field

Wilhelm Becker

Max-Born-Institut, Berlin, Germany

Workshop „Attoscience: Exploring and Controlling Matter on its Natural Time Scale“, KITPC, Beijing, May 12, 2011

Collaborators:

C. Figueira de Morisson Faria, University College, LondonS. P. Goreslavski, MEPhI, MoscowR. Kopold, Siemens, RegensburgX. Liu, CAS, WuhanD. B. Milosevic, U. SarajevoG. G. Paulus, U. JenaS. V. Popruzhenko, MEPhI, MoscowN. I. Shvetsov- Shilovski, U. Jena

MotivationNSDI knee experimentally measured for circular polarization

NSDI knee observed in completely classical (CC) andsemiclassical (tunneling-classical; TC) simulations forcircular polarization

Dependence of a process on ellipticity is indicative of themechanism

Nonsequential double ionization existsfor circular polarization

NSDI for a circularly polarized laser field

linear circular

G. D. Gillen, M. A. Walker, L. D. Van Woerkom, PRA 64, 043413 (2001)

magnesium, Ip1 = 7.6 eV, Ip2 = 15.0 eV, Ip3 = 80 eV, 120 fs, 800 nm

Double-ionization yield from completely classical (CC) simulations

X. Wang, J. H. Eberly, NJP 12, 093047 (2010)

Ip = 1.3 a.u.

Electron trajectories from completely classical double-ionization simulations

X. Wang, J. H. Eberly, NJP 12, 093047 (2010)

doubly-ionizing orbits tend to be „long orbits“

CC simulation: escape over the Stark saddle depends onparameters

helium, Ip = 2.24 a.u.a = 1 b = 1

no knee

magnesium, Ip = 0.83 a.u.a = 3 b = 1

a knee

F. Mauger, C. Chandre, T. Uzer, PRL 105, 083002 (2010)

Elliptical polarization helps revealing the mechanism

Ellipticity dependence reveals the mechanism

P. Dietrich, N. H. Burnett, M. Yu. Ivanov, P. B. Corkum, PRA 50, R3589 (1994)

HH 21 in argon, measuredand simulated

NSDI of argon, measuredand simulated

ellipticity

An example of ellipticity as a diagnostic toolNSDI of neon as a function of wavelength for various ellipticities

calculated by the tunneling-classical-trajectory model

transition to thestandard rescatteringmechanism at about

200 nm

constant intensityI = 1.0 x 1015 Wcm-2

X. Hao, G. Wang, X. Jia, W. Li, J. Liu, J. Chen, PRA 80, 023408 (2009)

Recollision and elliptical (linear --> circular) polarization

Simplest simple-man argument:for sufficiently large ellipticity, especially for circular polarization, an electron released with zero velocity will not return to its place of birth

no recollision-induced processes

However, electrons are released with nonzero distributionof transverse velocities

recollision is possible for suitable transversemomentum

(But, no HHG for circular polarization, QM dipole selection rule)

Quantum-orbit formalism

Formal description of recollision processes

|)'()(|)',(

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

)(|)(|)(

3

0

0

)1()0(

ttdttU

ttttUVtdtdt

tttdti

MMM

VolkovVolkovVolkov

Volkovfft

f

fififi

qqq

Er

Er

HATI into a state with final (drift) momentum p:Vf = continuum scattering potential), <f| = <p

Volkov |

= „direct“ + rescattered

1st-order Born approximation

Formal description of recollision processes

|)'()(|)',(

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

)(|)(|)(

3

0

0

)1()0(

ttdttU

ttttUTtdtdt

tttdti

MMM

VolkovVolkovVolkov

Volkovfft

f

fififi

qqq

Er

Er

HATI into a state with final (drift) momentum p:Vf = continuum scattering potential), <f| = <p

Volkov |

= „direct“ + rescattered

Low-frequency approximation (LFA)

)],',(exp[),',(' 3 qqq ttiSttgddtdtM fit

fi

Evaluation by stationary phase (steepest descent)with respect to the integration variables t, t‘, k

Saddle-point equations for high-order ATI

pItem

2))'((21 Aq

t

tdett

')()'( Aq

22 ))(())(( tete ApAq

0'/ tS

0/ tS

0/ qS

the (complex) solutions ts‘, ts, and qs (s=1,2,...) determine electron orbits in the laser field („quantum orbits“)

'))(())(( 2

'

2 tIededS Pt

tt

AqAp

pi Ite 2))(( 2 Aq

22 ))(())(( ff tete ApAq

f

i

ttif edtt )()( Aq

Saddle-point equations

elastic rescattering

f

i

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

return to the ion

tunneling at constant energy

Many returns: for given final state, there are many solutions of the saddle-point equations

„Long orbits“

Building up the ATIspectrum from quantum orbits

shortest two orbits 1+2

shortest six orbits 1 +...+ 6

shortest 14 orbits

Magnitude of the contributionsof the various pairs of orbits

Significance of longer orbitsdecreases due to spreading

x(t=ts‘) = 0, but Re [x(Re ts‘)] = „tunnel exit“ different from 0

Quantum orbits (real parts) for elliptical polarizationRe

y (a

. u.)

position of the ion tunnel exit

x = semimajor axisy = semiminor axis

Note:the shortest orbits require thelargest transverse momentato return

semimajor polarization axis

Why longer orbits require lower transverse momenta to return

short orbit: transverse drift is significant

Why longer orbits require lower transverse momenta to return

longer orbit: transverse drift is much reduced

The contribution of an orbit is weighted exponentially

prop. to exp(-pdrift2/p2)

short orbits have large pdrift and are suppressed

What is the difference between the saddle points forlinear and for elliptical polarization?

pItem

2))'((21 Aq

linear pol.: for Ip = 0 and qT = 0, the solution t‘ is real simple-man model

elliptical pol.: even for Ip = 0 and qT = 0,the solution t‘ is complex(cannot have both qx - eAx(t‘) = 0 and qy- eAy(t‘) = 0)

can only say that q - eA(t‘) is a complex null vector

Examples: HHG and HATI

Above-threshold ionization by an elliptically polarized laser field

= 0.5 = 1.59 eV

I = 5 x 1014 Wcm-2

R. Kopold, D. B. Milosevic, WB, PRL 84, 3831 (2000)

The plateaubecomes a stair

The shortest orbits make the smallest contributions,

but with the highestcutoff

Quantum orbits for elliptical polarization: Experiment vs. theory

The plateau becomesa staircase

= 0.36xenon at 0.77 x 1014Wcm-2

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

The shortest orbits arenot always the dominantorbits

Alternative description: quasienergy formalism (zero-range potential or effective-range theory)

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

N. L. Manakov, M. V. Frolov, B. Borca, A. F. Starace, JPB 36, R49 (2003)

A. V. Flegel, M. V. Frolov, N. L. Manakov, A. F. Starace, JPB 38, L27 (2005)

Staircase for HATI

= 0.5Ip = 0.9 eV

= 1.59 eV

I = 5 x 1014

Wcm-2

R. Kopold, D. B. Milosevic, WB, PRL 84, 3831 (2000)

Staircase for HHG

Ip = 0.9 eV

= 1.59 eV

I = 5 x 1014

Wcm-2

= 0.5

Quantum orbits in the complex t0 and t1 plane

Im

t 0t1

Reti

t0

orbits 1,2orbits 3,4orbits 5,6

HATI: * (asterisk)HHG: (diamond)

= 0.5, 780 nm, He5 x 1014 Wcm-2

HATI for various ellipticities

Ip = 0.9 eV

= 1.59 eV

I = 5 x 1014

Wcm-2

strong dropfor > 0.3

HHG for various ellipticities

D. B. Milosevic, JPB 33, 2479 (2000)

Ip = 13.6 eV

= 1 eV

I = 1.4 x 1014

Wcm-2

dramatic dropfor > 0.2

Cutoffs for HHG orbits

pp IcUcc 22

21max )2(1 21

pair of orbits1 c1 = 3.17 c2 = 1.32 (Lewenstein, Ivanov)

2 c1 = 1.54 c2 = 0.88

3 c1 = 2.40 c2 = 1.10

HATI cutoff

D. B. Milosevic, JPB 33, 2479 (2000)

pair 1 Emax = 10.01 Up + 0.54 IpM. Busuladzic, A. Gazibegovic-Busuladzic, D. B. Milosevic, Laser Phys. 16, 289 (2006)

Interference of direct and rescattered electrons

G. G. Paulus, F. Grasbon, A. Dreischuh,H. Walther, R. Kopold, WB, PRL 84, 3791 (2000)

experiment: 7.7 x 1013 Wcm-2

Xe 800 nm = 0.36theory: 5.7 x 1013 Wcm-2 „Xe (Ip = 0.436)“ = 0.48

Mechanism of the second plateau

rescattered

direct

The contributions of just the rescattered and just the direct electronsindividually are only smoothly dependent on the angle,

only the superposition is structured

Interference of direct and rescattered electrons

Conditions for interference between direct and rescattered electrons

energy

yiel

d

energy

yiel

ddirect

rescattered

direct

rescattered

for elliptical polarization, the yields of direct and rescatteredelectrons are comparable over a larger energy range

linear elliptical

See, however, Huismans et al., Science (2011)

Example: NSDI for elliptical polarization

NSDI from a simple semiclassical model

R(t) = ADK tunneling ratet = start time, t‘(t) = recollision timeE(t‘) = kinetic energy of the recolliding electron(t‘ - t)-3 = effect of spreadingVp1p2 = form factor (to be ignored)(...) = energy conservation in rescattering

C. Figueira de Morisson Faria, H. Schomerus, X. Liu, WB, PRA 69, 043405 (2004)

NSDI by an elliptically polarized field: the bad news

= 0 --> 0.48 o.o.m.!

N. I. Shvetsov-Shilovski, S. P. Goreslavski, S. V. Popruzhenko, WB, PRA 77, 063405 (2008)

Ti:Saneon

I = 8 x 1014

Wcm-2

NSDI for elliptical polarization: ion-momentum distribution

Ti:Sa neonI=8 x1014 Wcm-

2

first six returns

first return only

N. I. Shvetsov-Shilovski, S. P. Goreslavski, S. V. Popruzhenko, WB, PRA 77, 063405 (2008)

this case to be realizedby a single-cycle pulse

N. I. Shvetsov-Shilovski, S. P. Goreslavski, S. V. Popruzhenko, WB, PRA 77, 063405 (2008)

NSDI for elliptical polarization: electron-electron-momentum correlation

W(p1x,p2x|p1y>0,p2y>0)

first six returns

first return onlysingle-cycle pulse case!

Asymmetry of the momentum-momentum correlationbetween the first and the third quadrant

31

31wwww

8 x 1014 Wcm-2

4 x 1014 Wcm-2

asymmetry isstrongly intensity-dependentdepending upon which orbitsare dominant

= 10% for = 0.1yield is down by 3

should be measurable

Try some ellipticity

Coulomb focusing is desirable to increase the effects

ATI spectra for elliptical polarization are coming up fromX. Y. Lai and X. Liu