Tunneling ionization of atoms Christer Z. Bisgaard and Lars Bojer Madsen Citation: American Journal of Physics 72, 249 (2004); doi: 10.1119/1.1603274 View online: http://dx.doi.org/10.1119/1.1603274 View Table of Contents: http://scitation.aip.org/content/aapt/journal/ajp/72/2?ver=pdfcov Published by the American Association of Physics Teachers Articles you may be interested in Ionization and dissociation dynamics of vinyl bromide probed by femtosecond extreme ultraviolet transient absorption spectroscopy J. Chem. Phys. 140, 064311 (2014); 10.1063/1.4865128 Theoretical studies on tunneling ionizations of helium atom in intense laser fields J. Chem. Phys. 124, 144303 (2006); 10.1063/1.2183300 Ionization of Argon n =2 ( Ar +9 to Ar +16 ) by a “relativistic” laser field AIP Conf. Proc. 525, 667 (2000); 10.1063/1.1291983 Double ionization of a two-electron model atom in a single-cycle laser pulse AIP Conf. Proc. 525, 176 (2000); 10.1063/1.1291937 Calculating the Keldysh adiabaticity parameter for atomic, diatomic, and polyatomic molecules J. Chem. Phys. 108, 7739 (1998); 10.1063/1.476208 This article is copyrighted as indicated in the article. Reuse of AAPT content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 157.92.4.72 On: Tue, 19 Jan 2016 18:04:11 Tunneling ionization of atoms Christer Z. Bisgaard and Lars Bojer Madsena) Department of Physics and Astronomy, University of Aarhus, 8000 A˚ rhus C, Denmark ͑Received 24 February 2003; accepted 23 June 2003͒ We discuss the theory for the ionization of atoms by tunneling due to a strong external static electric field or an intense low frequency laser field. © 2004 American Association of Physics Teachers. ͓DOI: 10.1119/1.1603274͔ I. INTRODUCTION Z VϭϪ ϩFz. ͑1͒ r Strong field physics, that is, the interaction of strong fields with matter, is one of the current topics in atomic, molecular, ͑We shall use atomic units បϭeϭmϭ1 throughout unless and optical physics and brings together the newest laser tech- otherwise indicated.͒ The electron may tunnel through the nology and most advanced theoretical approaches. During barrier and escape along the negative z axis. The field the last ten years, developments in laser technology have strength and ionization potential in Fig. 1 are arbitrary but been in the direction of shorter and shorter pulses of higher representative values. The parameter ␬ introduced in the cap- 1 ␬ϭͱ and higher intensities. Pulses with a duration as short as a tion of Fig. 1 is defined as 2I p, where I p is the ioniza- few femtoseconds (1 fsϭ10Ϫ15 s) are now routinely pro- tion potential. duced, and peak intensities around 1014–1015 W/cm2 are not The tunneling ionization rate in a static, electric field was unusual.2 Note that if we combine the atomic units of energy derived by Landau and Lifshitz for the ground state of the ␣2 2 ␣ ␣ hydrogen atom,6 and later generalized to any asymptotic ( mec ), time (a0 / c), and length (a0), where is the Coulomb wave function by Smirnov and Chibisov.7 There fine structure constant, me is the mass of the electron, c is the speed of light, and a is the Bohr radius, we obtain the are some misprints in Ref. 7 which make the derivation dif- 0 ficult to follow and results from Ref. 7 should be taken with atomic unit of intensity I ϭ3.51ϫ1016 W/cm2, which is 0 caution. Perelomov, Popov, and Terent’ev8 corrected the re- within reach of experiment. The field strength corresponding sult from Ref. 7 and generalized it to electromagnetic fields ϭ to an intensity of I0 is 1 a.u. of field strength F0 5.14 of low frequency by taking the appropriate time average. It 9 ϫ10 V/cm. The new light sources are typically based on a was shown8,9 that a generalized theory, which also covers Ti:sapphire system operating at a wavelength of 800 nm cor- multiphoton processes, simplifies to the time-average of the ϳ 7 ␥ϭ␻ ␻ responding to a photon energy of 1.5 eV. Due to their high tunneling result when the Keldysh parameter, / t ,is intensities, other wavelengths can be produced by using non- ␻ much less than unity. The tunneling frequency t is defined linear optics. In this way femtosecond pulses covering the ␻ ϭ 1/2 ␻ ␻Ϫ1 by t F/(2I p) , and is the optical frequency; t is the spectrum from the infrared to the ultraviolet can be pro- tunneling time, that is, the time the electron spends under the duced. Such laser systems have recently been applied to barrier ͑see Fig. 1͒. study the behavior of atoms and molecules under intense 3 The requirement for the validity of the tunneling model for field ionization. an oscillating field is that the width of the barrier does not Although we will not discuss it here, we mention the phe- change during the time the electron spends traversing it, that nomenon of rescattering, which is one of the key ideas in is, the electron adiabatically follows the changes in the ex- strong field physics involving the tunneling ionization of ␥ 4 ternal field. In terms of these quantities, can be expressed atoms. To model this phenomenon, a three-step process is as considered: first the atom is ionized by the field, then the freed electron propagates in the electric field of the laser, and ␻ ␥ϭͱ2I . ͑2͒ finally it may be driven back to the parent ion. Here it may p F either scatter to produce high energy electrons, recombine with the emission of a high energy photon, or scatter inelas- We see that the requirement that the Keldysh parameter be tically, producing a doubly charged ion.5 In this three-step small is fulfilled when the intensity is high and the frequency rescattering model the ionization event is typically described is low, precisely the parameter range of present day intense by a tunneling formula. The ionization rate as a function of lasers. the phase of the field and the momentum distribution of the In this note we present a self-contained derivation of the liberated electrons serve as inputs for the propagation of the tunneling theory for ionization of atoms in laser fields. The electrons in the field. Hence, a good description of the initial derivation is a beautiful combination of physical insight and ionization process is needed in order to obtain accurate mod- technical manipulations, and it brings together many differ- els for whatever process the propagating electron subse- ent aspects of theoretical physics, including separation of quently undergoes. coordinates, asymptotic forms of wave functions, and semi- The ionization step is fairly simple because it involves the classical solutions. process of tunneling. As we shall return to shortly, the tun- neling picture is well-justified for intense low-frequency la- II. DERIVING THE IONIZATION RATE ser fields. Figure 1 shows the combined potential felt by the electron from an external electric field F along the z direc- The starting point in the derivation of the tunneling theory tion and from a nucleus of charge Z: for an atom in an external field is the observation that there 249 Am. J. Phys. 72 ͑2͒, February 2004 http://aapt.org/ajp © 2004 American Association of Physics Teachers 249 This article is copyrighted as indicated in the article. Reuse of AAPT content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 157.92.4.72 On: Tue, 19 Jan 2016 18:04:11 with ␰,␩෈͓0,ϱ͓ and ␾෈͓0,2␲͔. It follows that rϭ͑␰ϩ␩͒/2. We may express the Cartesian coordinates in terms of the parabolic ones as xϭͱ␰␩ cos ␾, yϭͱ␰␩ sin ␾, zϭ͑␰Ϫ␩͒/2. ͑4͒ The azimuthal angle ␾ is defined as in spherical coordinates. In parabolic coordinates the Laplace operator reads 2ץ 1 ץ ץ ץ ץ 4 ͒ ͑ 2ϭ ͫ ͩ ␰ ͪ ϩ ͩ ␩ ͪͬϩٌ ␾2 , 5ץ ␩ ␰␩ץ ␩ץ ␰ץ ␰ץ ␰ϩ␩ and the potential of Eq. ͑1͒ is 2Z 1 VϭϪ ϩ F͑␰Ϫ␩͒. ͑6͒ ␰ϩ␩ 2 Fig. 1. Full curve: the potential of Eq. ͑1͒ along the z axis with Zϭ1 and Ϫ 1ٌ2ϩ ␺ϭ ␺ If we write the Schro¨dinger equation, ( 2 V) E ,in Fϭ0.0169 a.u. This field strength corresponds to 8.69ϫ107 V/cm ͑1 a.u. of 9 parabolic coordinates, and express the wave function as a field strength is 5.14ϫ10 V/cm). Dashed curve: the pure Coulomb poten- ␺ tial. The horizontal line corresponds to ␬ϭ0.86, which is equivalent to a product of functions of each coordinate ϭ ␰ ␩ im␾ ͱ ␲ ionization potential of 10 eV. f 1( ) f 2( )e / 2 , with m the magnetic quantum number, we obtain 1 d df E␰ m2 F␰2 are regions in configuration space where the electron is suf- ͫ ͩ ␰ 1 ͪ ϩͩ Ϫ Ϫ ͪ ͑␰͒ͬ ͑␰͒ ␰ ␰ ␰ f 1 ficiently far away from the core that it effectively feels a f 1 d d 2 4 4 Coulomb attraction from the nucleus screened by all the 1 d df E␩ m2 F␩2 other electrons. We will show that the ionization rate can be ϩ ͫ ͩ ␩ 2 ͪ ϩͩ Ϫ ϩ ͪ ͑␩͒ͬ ͑␩͒ ␩ ␩ ␩ f 2 obtained for systems with one general property: the f 2 d d 2 4 4 asymptotic wave function must behave as the asymptotic ϭϪZ, ͑7͒ Coulomb wave function. The derivation involves the following steps. where we have multiplied through by Ϫ(␰ϩ␩)/2 and di- ␺ ͑a͒ Separate Schro¨dinger’s equation in parabolic coordi- vided by .