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