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The Zero-Range Potential in an Intense
Laser Field: Much more than a Toy
W. Becker1 and D. B. Milosevic1,2
1Max-Born-Institut,
2Faculty
Berlin, Germany
of Science, University of Sarajevo, Sarajevo,
Bosnia and Herzegovina
Quantum Optics VII, June 8 – 12, 2009, Zakopane, Poland
2 /  c  a 2
c  
An effective-source model for
above-threshold ionization
Volkov solution
regularized ZRP in an arbitrary number of spatial dimensions
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
in any number of dimensions but one, the ZRP d(x) must
be regularized to allow for a normalizable bound state:
d(x)  dreg(x)
Alternatively, a separable potential V = l|a> <a| has a single bound
state for appropriate l
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 = -2p exp[-2p/a2
+ y(1)]/L2
Many applications in one and three dimensions
neutron-proton scattering (Fermi 1936)
nuclear physics
solid state Kronig-Penney
BE condensates with attractive short-range interaction
modeling the tip in scanning-tunneling microscopy
electrons bound in negative ions
laser-atom physics
large wave length (s-wave) scattering
very 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
t
 (r, t )   dt ' d r ' G
3
Volkov
(rt ; r ' t ' )V (r ' ) (r ' t ' )

The wave function within the range of V(r) determines
the wave function everywhere
(r' t ' ) y 0 (r' ) exp(i | E0 | t ' )
„direct ionization“ (no modification of the ground-state
wave function by the laser field)
(virtually exact for a circularly polarized field)
Evolution of the atomic ground state in a laser field
propagation in the laser field only, no potential
t
 (r, t )   dt ' d r ' G
3
Volkov
(rt ; r ' t ' )V (r ' ) (r ' t ' )

The wave function within the range of V(r) determines
the wave function everywhere
(r' t ' ) y 0 (r' ) exp(i | E0 | t ' )
„direct ionization“ (no modification of the ground-state
wave function by the laser field)
(virtually exact for a circularly polarized field)
Evolution of the atomic ground state in a laser field
t
 (r, t )   dt ' d r ' G
3
Volkov
(rt ; r ' t ' )V (r ' ) (r ' t ' )

The wave function within the range of V(r) determines
the wave function everywhere
Insert the integral equation into itself:
t
 (r, t )   dt ' d 3r ' GVolkov (rt ; r ' t ' )V (r ' ) 

t'
  dt ' ' d 3r ' ' GVolkov (r ' t ' , r ' ' t ' ' )V (r ' ' )y 0 (r ' ' ) exp(i | E0 | t ' ' )

allows for a modification of the wave function within the range
of 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, w = 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
1.0 I0
0.5 I0
intensity
increases
by 6%
Paulus, Grasbon, Walther, Kopold, Becker,
PRA 64, 021401 (2001)
see, also, Hansch, Walker, van Woerkom, PRA 55, R2535 (1997)
Hertlein, Bucksbaum, Muller, JPB 30, L197 (1997)
S-matrix element for ionization from the ground state
|y0> into a continuum state |yp> with momentum p

t


M p   dt f  dti y p(Volkov ) (t f ) | VU (Volkov ) (t f , ti )V | y 0 (ti )
i
|yp(t) > = Volkov state, U(Volkov)(t,t‘) = Volkov propagator
V = binding potential
U
(Volkov )
(t , t ' )   d ky k (t )y k (t ' )
3
Saddle-point (steepest-descent) evaluation of the amplitude
t

i
3
2
M p   dt f  dti  d km p (k , t f , ti ) exp  d (p  eA( )) 
 2  




t
f
f
 it

2
 exp  d (k  eA( ))  I pti 
 2 t

f
i
Find values of k, tf, and ti, so that the exponentials be stationary:
 / k...   / t f ...   / ti ...  0
(k  eA(ti ))2  2 I p
(k  eA(t f ))2  (p  eA(t f ))2
k (t f  ti )  tt deA( )
f
i
saddle-points equs. with infinitely many (complex) solutions ks, tfs, tis (s=1,2,...)
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)
Saddle-point equations
(k  eA(ti ))2  2 I p
tunneling at constant energy
k (t f  ti )  tt deA( )
f
i
return to the ion
k (t
f

ti
)

t

i
t
(k  eA(t f ))2  (p  eA(t f ))2
f

d

eA (
)
elastic rescattering
Quantum-orbit expansion of the ionization amplitude
M p   mp (k s , t fs , tis ) exp iSp (k s , t fs , tis )
s
coherent superposition of different pathways into the same final state
+
+
+
realization of Feynman‘s path integral
each orbit by itself depends only very smoothly on intensity
NB: coherent-superposition effects are quantum effects
Salieres et al., Science 292, 902 (2001)
+....
Enhancements: SFA-type theory vs experiment
Focal-averaged zero-range potential
SFA simulation
argon spectra,
6.45 1013 Wcm-2 < I < 6.88 1013 Wcm-2
Hertlein, Bucksbaum, and Muller, JPB 30, L197 (1997)
6.39 1013 Wcm-2 < I < 6.91 1013 Wcm-2
Kopold, Becker, Kleber, Paulus,
JPB 38, 217 (2002)
Physical origin of the enhancement
Electron energies:
Ep = p2/(2m) = (N+n) w - Ip - Up
At the channel-closing intensity:
Up + Ip = Nw,
Electrons are emitted with zero drift momentum, p = 0 (n=0).
(At channel closings: Ep = nw)
Many recurrences: many
opportunities for rescattering
Constructive interference
of long quantum orbits
Quantum effect!!!
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 quantum
orbits included in the
calculation
a few orbits are
sufficient to
reproduce the
spectrum,
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 quantum
orbits included in the
calculation
a few orbits are
sufficient to
reproduce the
spectrum,
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
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
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 qr
pfT = p sin qr
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 qr
pfT = p sin qr
M. Okunishi, T.Morishita, G. Pruemper, K. Shimada, C. D. Lin, S. Watanabe, K. Ueda, PRL 100, 143001 (2008)
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
M fi  M fi (0)  M fi (1)
= „direct“ + rescattered

 i  dt y f (t ) | r  E(t ) | y 0 (t )


t
  dt  dt 'y f (t ) | V f UVolkov (t , t ' )r  E(t ' ) | y 0 (t ' )
 
UVolkov (t , t ' )   d 3q | y qVolkov (t )y qVolkov (t ' ) |
Formal description of recollision processes
M fi  M fi (0)  M fi (1)

 i  dt y f (t ) | r  E(t ) | y 0 (t )
Low-frequency
approximation
LFA


t
  dt  dt 'y f (t ) | T f UVolkov (t , t ' )r  E(t ' ) | y 0 (t ' )
 
UVolkov (t , t ' )   d 3q | y qVolkov (t )y qVolkov (t ' ) |
going beyond the first-order Born approximation
Vf  Tf
T f  V f  V f GVf V f
A. Cerkic, E. Hasovic, D.B. Milosevic, WB, PRA, 79, 033413 (2009)
First-order Born vs. Low-Frequency Approximation
Ar
2.3x1014 Wcm-2
800nm
LFA generates zeros in the differential cross section
Comparing the calculated electron-argon+ cross section
with the cross section extracted from HATI calculations
cross
section
s
calculated
extracted from HATI
Comparison of first-order Born
vs Low-Frequency approximation
High-order above-threshold
ionization
1BA
in the momentum (px,pz) plane
xenon at 1.5x1014 Wcm-2 760 nm
laser pol.direction
1BA is only sufficient
(if at all) in the direction
of the laser polarization
LFA
Conclusion
The zero-range potential provides a perfect model for
the laser-atom interaction
Improvements allow for an atom-specific quantitative
description of ionization spectra
Thank you, Krzysztof, for very many years
of friendship and inspiration