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, London
S. P. Goreslavski, MEPhI, Moscow
R. Kopold, Siemens, Regensburg
X. Liu, CAS, Wuhan
D. B. Milosevic, U. Sarajevo
G. G. Paulus, U. Jena
S. V. Popruzhenko, MEPhI, Moscow
N. I. Shvetsov- Shilovski, U. Jena
Motivation
NSDI knee experimentally measured for circular polarization
NSDI knee observed in completely classical (CC) and
semiclassical (tunneling-classical; TC) simulations for
circular polarization
Dependence of a process on ellipticity is indicative of the
mechanism
Nonsequential double ionization exists
for circular polarization
NSDI for a circularly polarized laser field
magnesium, Ip1 = 7.6 eV, Ip2 = 15.0 eV, Ip3 = 80 eV, 120 fs, 800 nm
linear
circular
G. D. Gillen, M. A. Walker, L. D. Van Woerkom, PRA 64, 043413 (2001)
Double-ionization yield from completely classical (CC) simulations
Ip = 1.3 a.u.
X. Wang, J. H. Eberly, NJP 12, 093047 (2010)
Electron trajectories from completely classical
double-ionization simulations
doubly-ionizing orbits tend to be „long orbits“
X. Wang, J. H. Eberly, NJP 12, 093047 (2010)
CC simulation: escape over the Stark saddle depends on
parameters
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
HH 21 in argon, measured
and simulated
NSDI of argon, measured
and simulated
ellipticity
P. Dietrich, N. H. Burnett, M. Yu. Ivanov, P. B. Corkum, PRA 50, R3589 (1994)
An example of ellipticity as a diagnostic tool
NSDI of neon as a function of wavelength for various ellipticities
calculated by the tunneling-classical-trajectory model
constant intensity
I = 1.0 x 1015 Wcm-2
transition to the
standard rescattering
mechanism at about
200 nm
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 distribution
of transverse velocities
recollision is possible for suitable transverse
momentum
(But, no HHG for circular polarization, QM dipole selection rule)
Quantum-orbit formalism
Formal description of recollision processes
M fi  M fi (0)  M fi (1)
= „direct“ + rescattered

 i  dt  f (t ) | r  E(t ) |  0 (t )


t
1st-order Born approximation
  dt  dt ' f (t ) | V f UVolkov (t , t ' )r  E(t ' ) |  0 (t ' )
 
UVolkov (t , t ' )   d 3q |  qVolkov (t ) qVolkov (t ' ) |
HATI into a state with final (drift) momentum p:
Vf = continuum scattering potential), <f| = <pVolkov |
Formal description of recollision processes
M fi  M fi (0)  M fi (1)
= „direct“ + rescattered

 i  dt  f (t ) | r  E(t ) |  0 (t )


t
Low-frequency approximation (LFA)
  dt  dt ' f (t ) | T f UVolkov (t , t ' )r  E(t ' ) |  0 (t ' )
 
UVolkov (t , t ' )   d 3q |  qVolkov (t ) qVolkov (t ' ) |
HATI into a state with final (drift) momentum p:
Vf = continuum scattering potential), <f| = <pVolkov |

t
M fi   dt  dt ' d 3qg (t , t ' , q) exp[ iS fi (t , t ' , q)]
 
Evaluation by stationary phase (steepest descent)
with respect to the integration variables t, t‘, k
Saddle-point equations for high-order ATI

t
S   d (p  eA( ))   d (q  eA( ))  I Pt '
t
S / t '  0
S / q  0
S / t  0
2
2
t'
1
(q  eA(t ' )) 2   I p
2m
t
(t  t ' )q  e  dA( )
t'
2
(q  eA(t ))  (p  eA(t ))2
the (complex) solutions ts‘, ts, and qs (s=1,2,...) determine
electron orbits in the laser field („quantum orbits“)
Saddle-point equations
(q  eA(ti )) 2  2 I p
tunneling at constant energy
q(t f  ti )  tt deA( )
f
i
return to the ion
k (t
f

ti )

t

i
t
(q  eA(t f )) 2  (p  eA(t f )) 2
f

d

eA (
)
elastic rescattering
Many returns: for given final state, there are
many solutions of the saddle-point equations
„Long orbits“
Building up the ATI
spectrum from quantum
orbits
shortest 14 orbits
shortest six orbits 1 +...+ 6
shortest two orbits 1+2
Magnitude of the contributions
of the various pairs of orbits
Significance of longer orbits
decreases due to spreading
x(t=ts‘) = 0, but Re [x(Re ts‘)] = „tunnel exit“ different from 0
Quantum orbits (real parts) for elliptical polarization
position of the ion
tunnel exit
x = semimajor axis
y = semiminor axis
Note:
the shortest orbits require the
largest transverse momenta
to 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/Dp2)
short orbits have large pdrift and are suppressed
What is the difference between the saddle points for
linear and for elliptical polarization?
1
(q  eA(t ' )) 2   I p
2m
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
x = 0.5
w = 1.59 eV
I = 5 x 1014 Wcm-2
The plateau
becomes a stair
The shortest
orbits make
the smallest
contributions,
but with the highest
cutoff
R. Kopold, D. B. Milosevic, WB, PRL 84, 3831 (2000)
Quantum orbits for elliptical polarization:
Experiment vs. theory
x = 0.36
xenon at 0.77 x 1014Wcm-2
The plateau becomes
a staircase
The shortest orbits are
not always the dominant
orbits
Salieres, Carre, Le Deroff, Grasbon, Paulus, Walther, Kopold, Becker, Milosevic, Sanpera, Lewenstein,
Science 292, 902 (2001)
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
x = 0.5
Ip = 0.9 eV
w = 1.59 eV
I = 5 x 1014
Wcm-2
R. Kopold, D. B. Milosevic, WB, PRL 84, 3831 (2000)
Staircase for HHG
x = 0.5
Ip = 0.9 eV
w = 1.59 eV
I = 5 x 1014
Wcm-2
Quantum orbits in the complex t0 and t1 plane
Re wti
orbits 1,2
orbits 3,4
orbits 5,6
HATI: * (asterisk)
HHG: (diamond)
wt0
wt1
x = 0.5, 780 nm, He
5 x 1014 Wcm-2
HATI for various ellipticities
Ip = 0.9 eV
w = 1.59 eV
I = 5 x 1014
Wcm-2
strong drop
for e > 0.3
HHG for various ellipticities
Ip = 13.6 eV
w = 1 eV
I = 1.4 x 1014
Wcm-2
dramatic drop
for e > 0.2
D. B. Milosevic, JPB 33, 2479 (2000)
Cutoffs for HHG orbits


 max  c1 1  (2  c2 ) U p  c2 I p
pair of orbits
2
1
c1 = 3.17 c2 = 1.32
2
c1 = 1.54 c2 = 0.88
3
c1 = 2.40 c2 = 1.10
x

1 x 2
(Lewenstein, Ivanov)
D. B. Milosevic, JPB 33, 2479 (2000)
HATI cutoff
pair 1
Emax = 10.01 Up + 0.54 Ip
M. 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)
Mechanism of the second plateau
theory: 5.7 x 1013 Wcm-2
„Xe (Ip = 0.436)“ x = 0.48
experiment: 7.7 x 1013 Wcm-2
Xe 800 nm x = 0.36
Interference of direct and rescattered electrons
rescattered
direct
The contributions of just the rescattered and just the direct electrons
individually are only smoothly dependent on the angle,
only the superposition is structured
Conditions for interference between direct
and rescattered electrons
direct
linear
direct
rescattered
energy
elliptical
rescattered
energy
for elliptical polarization, the yields of direct and rescattered
electrons are comparable over a larger energy range
See, however, Huismans et al., Science (2011)
Example: NSDI for elliptical polarization
NSDI from a simple semiclassical model
R(t) = ADK tunneling rate
t = start time, t‘(t) = recollision time
E(t‘) = kinetic energy of the recolliding electron
(t‘ - t)-3 = effect of spreading
Vp1p2 = form factor (to be ignored)
d(...) = 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
x = 0 --> 0.4
8 o.o.m.!
Ti:Sa
neon
I = 8 x 1014
Wcm-2
N. I. Shvetsov-Shilovski, S. P. Goreslavski, S. V. Popruzhenko, WB, PRA 77, 063405 (2008)
NSDI for elliptical polarization: ion-momentum distribution
Ti:Sa neon
I=8 x1014 Wcm-2
first six returns
first return only
this case to be realized
by 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 only
single-cycle pulse case!
N. I. Shvetsov-Shilovski, S. P. Goreslavski, S. V. Popruzhenko, WB, PRA 77, 063405 (2008)
Asymmetry of the momentum-momentum correlation
between the first and the third quadrant
w1  w3

w1  w3
asymmetry is
strongly
intensity-dependent
depending upon
which orbits
are dominant
 = 10% for x = 0.1
yield is down by 3
should be measurable
8 x 1014 Wcm-2
4 x 1014 Wcm-2
Try some ellipticity
Coulomb focusing is desirable to increase the effects
ATI spectra for elliptical polarization are coming up from
X. Y. Lai and X. Liu