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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 deA( ) 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 deA( ) 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