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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
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c2/2 ac
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An effective-source model for above-threshold ionization
Volkov solution
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regularized ZRP in an arbitrary number of spatial dimensions
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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
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regularized ZRP in an arbitrary number of spatial dimensions
E0 = - ma2/2
d = 5
d = 2 E0 = -2 exp[-2/a2
+ (1)]/L2
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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)
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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))
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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)
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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)
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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
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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)
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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
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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)
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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
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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)
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Intensity-dependent enhancements of groups of above-threshold-ionization peaks in the
rescattering regime
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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
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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
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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)
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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
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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
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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
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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)
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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)
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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
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Constructive interference of many long orbits
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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
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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)
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Quantitative rescattering theory:
Reconstruction of the electron-ion potential
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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?
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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?
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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)
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the same for xenon
theory: effective model potential, Coulomb + short range
M. Okunishi et al., PRL 100, 143001 (2008)
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Formal description of recollision processes
|)'()(|)',(
)'(|)'()',(|)('
)(|)(|)(
3
0
0
)1()0(
ttdttU
ttttUVtdtdt
tttdti
MMM
VolkovVolkovVolkov
Volkovff
t
f
fififi
qqq
Er
Er
= „direct“ + rescattered
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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
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First-order Born vs. Low-Frequency Approximation
LFA generates zeros in the differential cross section
Ar
2.3x1014 Wcm-2
800nm
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crosssection
Comparing the calculated electron-argon+ cross sectionwith the cross section extracted from HATI calculations
calculated extracted from HATI
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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
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Conclusion
The zero-range potential provides a perfect model forthe laser-atom interaction
Improvements allow for an atom-specific quantitativedescription of ionization spectra
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Thank you, Krzysztof, for very many yearsof friendship and inspiration
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