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
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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 ͓0,2͔. It follows that rϭ͑ϩ͒/2. We may express the Cartesian coordinates in terms of the parabolic ones as
xϭͱ cos , yϭͱ sin , zϭ͑Ϫ͒/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 F͑Ϫ͒. ͑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 F2 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 F2 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 . ϭ ͱ nates in the presence of an external field. If we introduce f 1( ) 1( )/ and f 2( ) ϭ ͱ ͑b͒ Find the asymptotic solution to the pure Coulomb 2( )/ , the differential operators simplify, and we are problem. led to ͑c͒ Combine these two results to obtain an expression for 2 the ionization rate which only requires the knowledge 1 d 1 E Ϫ ϩU ͑͒ ϭ , ͑8a͒ of the wave function on the outside of the barrier. 2 d2 1 1 4 1 ͑d͒ Obtain a semiclassical expression for the wave function 2 outside the barrier and in the classically forbidden re- 1 d 2 E Ϫ ϩU ͑͒ ϭ , ͑8b͒ gion under the barrier. 2 d2 2 2 4 2 ͑e͒ Write the rate in terms of the semiclassical wave func- tion outside the barrier and find an expression that de- with the constraint pends only on the normalization of the wave function.  ϩ ϭ ͑ ͒ This normalization is determined by requiring the 1 2 Z. 9 semiclassical wave function in a certain region inside the barrier to match the asymptotic Coulomb wave The effective one-dimensional potentials are function. The rate is then described completely by pa-  m2Ϫ1 F rameters characterizing the asymptotic Coulomb wave ͑͒ϭϪ 1 ϩ ϩ ͑ ͒ U1 2 , 10a function. 2 8 8 ͑ ͒ f Perform the appropriate time average to obtain the rate  2Ϫ 2 m 1 F in a low frequency laser field. U ͑͒ϭϪ ϩ Ϫ . ͑10b͒ 2 2 82 8 In a teaching context we suggest that the lecturer provide an overview of the derivation and leave some parts of the As two generic examples, we have sketched these potentials derivation as student exercises. As will become clear, any of in Fig. 2 for mϭ0,2. the points in Secs. II A and II B may be formulated as prob- As seen from the potential of Eq. ͑1͒, the ionization will lems. occur in the Ϫz direction, which in parabolic coordinates is along the coordinate ͓see Eq. ͑3͔͒. The derivation of the results ͑8͒ and ͑10͒ may be formu- A. Separation in parabolic coordinates: Static field lated as a problem. For example, the opening question could be ͑a͒ obtain V in parabolic coordinates. Equation ͑5͒ could Parabolic coordinates are defined as6 be provided by the teacher, and part ͑b͒ could be to perform ϭrϩz, ϭrϪz, ϭarctan͑y/x͒, ͑3͒ the separation of coordinates.
250 Am. J. Phys., Vol. 72, No. 2, February 2004 C. Z. Bisgaard and L. B. Madsen 250 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 Fig. 2. Full curves: the potentials ͑a͒ ϭ ͑ ͒ ϭ ͑ ͒ U2( ), m 0; b U1( ), m 0; c ϭ ͑ ͒ ϭ U2( ), m 2; and d U1( ), m 2.  The separation constant 2 has been chosen according to Eq. ͑19͒. For all the plots the parameters are Zϭ1, ϭ0.86, and Fϭ0.0169, as in Fig. 1. Dashed curves: the pure Coulomb po- tentials. The horizontal line corre- sponds to the effective binding energy Ϫ in the and coordinates, I p /4.
B. Asymptotic Coulomb wave function and separation The expansion in the Ϫz direction has the same magnitude, constants but the sign should be changed according to Y lm( Ϸ0,)/Y (Ϸ,)ϭ(Ϫ1)lϪ͉m͉. The sign of the spherical Finding the asymptotic wave function in spherical coordi- lm nates may be formulated as a brief problem. harmonics turns out to be unimportant because Q(l,m) only ͉ ͉2 Problem: Show that the asymptotic form ͑large r) of the enters the final expression for the ionization rate as Q . ϷϪ Coulomb wave function with energy EϭϪ2/2 is given by In the region of interest z is large and negative (r z). In parabolic coordinates this region is equivalent to ӷ. ϭ ͒ ͒ϭ ͑Z/͒ Ϫ1 Ϫr ͒ ͑ ͒ C R͑r Y lm͑ , Dr e Y lm͑ , . 11 We expand the left-hand side of the equality cos Ј ϭϪ͑Ϫ͒/͑ϩ͒ to lowest order in Ј and the right-hand Solution: Substitute the expression of R(r) from Eq. ͑11͒ side to lowest order in / and find that into the radial equation6 1 d dR l͑lϩ1͒ sin ϭsin ЈϷЈϷ2ͱ/, ͑15͒ ͩ 2 ͪ Ϫ ϩ ͑ Ϫ ͒ ϭ ͑ ͒ 2 r 2 R 2 E V R 0. 12 r dr dr r where ͑Ј͒ is the angle with the positive ͑negative͒ z axis. For VϭϪZ/r it is then readily verified that R(r) is the cor- We may thus write the asymptotic wave function as rect solution to leading order in 1/r. 2Ϫ ͑Z/͒ ϩ1 eim In Eq. ͑11͒ D is a normalization constant that is only Ϸ ͉m͉/2 Ϫ/2Z/ Ϫ ͉m͉/2 Ϫ1 Ϫ/2 C B e e , known in analytical form for the pure Coulomb problem ͉m͉! ͱ2 10 ͑see, for example, Ref. 6͒. Ammosov, Delone, and Krainov ͑16͒ obtained a general, analytical expression in terms of effective quantum numbers by considering the semiclassical solutions where we have introduced to the radial wave equation. ϵ ͑ ͒ ͑ ͒ Ionization occurs near the axis of the field, that is, along B Q l,m D, 17 the negative z axis. We leave it as an exercise to verify the for the asymptotic constant. In general, D should be calcu- following approximation for spherical harmonics close to the lated numerically by matching the solution of the Schro¨- ϩ ͑Ϸ ͒ z direction 0 : dinger equation to the form ͑11͒.   ͓ ͑ ͔͒ sin͉m͉ eim The separation constants, 1 and 2 see Eq. 9 are de- ͑ ͒Ϸ ͑ ͒ ͑ ͒ termined by requiring the asymptotic solution ͑16͒ to solve Y lm , Q l,m ͉m͉͉ ͉ ͱ , 13 2 m ! 2 Eq. ͑8͒ in the limit of vanishing field strength F→0. This 3 with approach is valid for F/ Ӷ1 ͑see Ref. 6, p. 292͒. The in- equality F/3Ӷ1 will also be used in Sec. II F where an 2lϩ1 ͑lϩ͉m͉͒! Q͑l,m͒ϵ͑Ϫ1͒(mϩ͉m͉)/2ͱ . ͑14͒ expression for the rate in an oscillating field is obtained. 2 ͑lϪ͉m͉͒! From Eq. ͑16͒ we read off the -dependence:
251 Am. J. Phys., Vol. 72, No. 2, February 2004 C. Z. Bisgaard and L. B. Madsen 251 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 2  2Ϫ 1/2 ͑͒ 2 m 1 1 1 ͉m͉/2 Ϫ/2 ϭͫϪͩ ͪ ϩ Ϫ ϩ ͬ ͑ ͒ ϭ e . ͑18͒ p 2 F , 27 ͱ 2 4 4 We leave it as an exercise to substitute Eq. ͑18͒ into Eq. ͑8a͒ which shows that Eq. ͑26͒ is fulfilled in the ranges to obtain  2 2 2 ӶӶ ӷ ͑ ͒ 2 , . 28  ϭ ͉͑m͉ϩ1͒, ͑19͒ F F 1 2 It is seen from Eq. ͑26͒ that the semiclassical solutions do  ͑ ͒ which fixes 2 through Eq. 9 . not apply near the classical turning points (pϭ0). For our purposes the interesting turning point is the one on the outer side of the barrier, which from Eq. ͑27͒ for ӷ1 is seen to be C. Expression for the ionization rate Ϸ2 ͑ ͒ 0 /F. 29 In Sec. II A we found that the wave function could be written as Therefore, when relating the semiclassical solutions in the Ͼ classically allowed ( 0) and forbidden regions ( ͒ ͒ im 1͑ 2͑ e Ͻ ), care should be taken around Ϸ . ϭ . ͑20͒ 0 0 ͱ ͱ ͱ2 The asymptotic Coulomb wave function is an exponen- tially decreasing function of . Therefore, the semiclassical The ionization rate is found by integrating the probability wave function in the classically forbidden region is of the current density in the z direction, jz , over a surface orthogo- form nal to it C ͑͒ϭ expͩ Ϫ ͵ ͉p͉dͪ ͑Ӷ ͒. ͑30͒ 2 0 ϭϪ͵ ͑ ͒ ͱ W jz dS. 21 ͉p͉ 0 The probability current density for z large and negative (z Equation ͑30͒ must be related to the semiclassical solution Ͼ ϷϪ/2) is in the classically allowed region, 0 representing a running-wave solution * ͉ ͉2 d dץ *ץ i ϵ ͩ Ϫ ͪ Ϸ 1 ͩ * 2 Ϫ 2 ͪ i 2 2 . C ץ * ץ jz 2 z z 2 d d ͑͒ϭ 1 ͩ ͪ ͑Ͼ ͒ ͑ ͒ ͑ ͒ 2 exp i ͵ pd 0 . 31 22 ͱp For ionization near the z axis the differential surface element is written as To fix the relation between the exponents and the normaliza- tion constants C and C1 , we may match directly to the quan- 1 1 tum mechanical solutions in the two regions or, as will be dSϭ d dϭͱd ͱ dϷ d d, ͑23͒ 2 2 done here, use purely semiclassical methods. To this end, the first step is to consider the semiclassical wave function in a so the rate can be expressed as Ͻ ͉Ϫ ͉ region 0 with 0 small enough that we can make ϱ ͉ ͉͒2 a Taylor expansion of the potential around ϭ : i d 2* d 2 1͑ 0 Wϭ ͫ Ϫ* ͬ ͵ d . ͑24͒ 2 2 d 2 d 0 dU2 U ͑͒ϷU ͑ ͒ϩ͑Ϫ ͒ ͯ . ͑32͒ ͑ ͒ 2 2 0 0 For 1( ) we use Eq. 18 , and express the rate in terms of d the part of the wave function: 0 ͉ ͉ Note that this expansion is not necessarily in contradiction i m ! d 2* d 2 Wϭ ͫ Ϫ* ͬ. ͑25͒ with the requirement that should be far enough away from ͉m͉ϩ1 2 2 2 d d 0 for the semiclassical solutions to apply. If we use the ͑ ͒ Problem: Why is it accurate to use Eq. ͑18͒ for ()in approximation in Eq. 32 , the momentum becomes p 1 Ϸ 1/2 Ϫ 1/2 ϭϪ the evaluation of the integral in Eq. ͑24͒? (2F0) ( 0) , where F0 dU2 /d , and we find 2 ͵ ͉ ͉ ϭ ͑ ͒1/2͑ Ϫ͒3/2 ͑ ͒ p d 2F0 0 . 33 3 D. Semiclassical solutions 0 To obtain a closed, analytic expression for the rate, the To get around the classical turning point, we apply an analytical continuation of the wave function into the com- semiclassical solutions for the function () are needed 2 plex plane. This continuation allows us to get from the under and outside the barrier. Such solutions may be applied classically forbidden to the classically allowed region if6 through a semicircle in the complex plane, and it enables 1 dU2 us to connect the solutions through a region where both the ͯ ͯӶ1, ͑26͒ ͑ ͒ p3 d Taylor expansion and the semiclassical solutions formally apply. We note that for Ͻ , the momentum is purely ͑ ͒ 0 where the potential U2( ) is given by Eq. 10b . The imaginary, giving ͉p͉ϭϪip. If we introduce momentum corresponding to this potential follows from ͑ ͒ Ϫ ϭ i ͑ ͒ Eq. 8b , 0 e , 34
252 Am. J. Phys., Vol. 72, No. 2, February 2004 C. Z. Bisgaard and L. B. Madsen 252 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 and use Eq. ͑33͒, we may express the semiclassical wave FЈ 1/2 function as ͩ 1Ϫ ͪ Ϫ1  dЈ  2 ͵ 2 ϭ 2 2 1/2 ln 1/2 ͯ C ͩ ͪ ͑ Ϫ Ј͒ Ј Ј ͑͒ϭ 0 F F 2 i(ϩ) 1/4 ͩ 1Ϫ ͪ ϩ1 ͓2F0 e ͔ 2 0 2 1/2 3/2 i͑3/2͒(ϩ) ϫexp͑Ϫ ͑2F ͒ e ͒. ͑35͒  3 0 F 2 / Ϸlnͩ ͪ . ͑43͒ Passing from the classically forbidden to the classically al- 42 lowed region then corresponds to letting the angle in the Ϸ2 ͓ ͑ ͔͒ complex plane undergo the change ϭ→0, so the semi- In these evaluations we used that 0 /F see Eq. 29 . Ͼ Finally, the semiclassical wave function corresponding to the classical wave function for 0 becomes one-dimensional Schro¨dinger equation in the coordinate is C 2 ͑͒ϭ ͩ Ϫ ͑ ͒1/23/2 i͑3/2͒ ͪ approximately exp 2F e . 2 ͓ i ͔1/4 0  2F0 e 3 1/2 2 / 2 F 3 ͑36͒ ͑͒ϷCͩ ͪ ͩ ͪ eϪ/2e /3F. ͑44͒ 2 42 If we write this result as If we substitute Eqs. ͑18͒ and ͑44͒ into Eq. ͑20͒, the total CeϪi/4 2 ͑͒ϭ ͩ ͑ ͒1/23/2ͪ ͑ ͒ semiclassical wave function becomes 2 1/4 exp i 2F0 , 37 ͓2F0 ͔ 3 Z/ Ϫ͉m͉/2 Ϫ 1/2 Ϫ Ϫ ϩ ͉ ͉ ϩ F ͑ ͒ ϭC 1/22 Z/ m /2 1ͩ ͪ and take it as the Taylor expansion of the general form 31 , 22 we fix C1 in terms of C, and obtain im 3 e C ϫe /3F͉m͉/2eϪ/2Z/ Ϫ ͉m͉/2 Ϫ1eϪ/2 . ͑45͒ ͑͒ϭ ͩ ͵ Ϫ ͪ ͑Ͼ ͒ ͑ ͒ ͱ 2 exp i pd i 0 . 38 2 ͱp 4 If we compare this expression for the asymptotic Coulomb ͑ ͒ E. Putting it all together wave function in parabolic coordinates given in Eq. 16 ,we find We are now ready to put all the pieces together to obtain 2 Z/ Ϫ͉m͉/2 Ϫ 1/2 B 2 3 the final expression for the ionization rate. We use Eq. ͑38͒ Cϭ1/2 ͩ ͪ eϪ /3F, ͑46͒ 2͉m͉/2͉m͉! F and neglect variations in p in the prefactor and find and the ionization rate can hereby be expressed in terms of d 2 C ͓ ͑ ͔͒ ϭ ip expͩ i ͵ pdϪi ͪ . ͑39͒ the parameters B Eq. 17 , , and m from the asymptotic d ͱp 4 Coulomb wave function as ͑ ͒ ͑ ͒ ͉B͉2 1 23 2Z/ Ϫ͉m͉Ϫ1 If we substitute Eq. 39 into Eq. 25 , the rate becomes ϭ ͩ ͪ Ϫ23/3F ͑ ͒ W ͉m͉͉ ͉ 2Z/ Ϫ1 e . 47 ͉m͉! 2 m ! F ϭ ͉ ͉2 ͑ ͒ W ͉m͉ϩ1 C . 40 Equation ͑47͒ is exactly the expression stated in Ref. 8. This expression is very general. For a given nuclear charge The semiclassical wave function under the barrier is given by ϭ2 ͑ ͒ ͑ ͒ Z, ionization potential I p /2, field strength F, and mag- Eq. 30 with the momentum given in Eq. 27 . In the com- netic quantum number m, the only unknown in Eq. ͑47͒ is plete range of integration we have ӷ1, and at the left end the constant B. This constant in turn is determined by the point ͑where we will match to the asymptotic Coulomb wave analytical function Q(l,m) from Eq. ͑14͒ and the normaliza- function͒ we further use the approximation necessary for the semiclassical wave function to be valid, that is, Eq. ͑28͒.To tion constant D, determined by matching the quantum me- obtain the wave function we therefore introduce some ap- chanical wave function of the atom to the asymptotic Cou- ͑ ͒ proximations. In the prefactor all terms depending on will lomb form 11 . 3 be neglected according to Eq. ͑28͒, whereas in the integrand Usually F/2 is much smaller than unity, and hence it Ϫ ͑ ͒ in the exponent the term proportional to 2 will be ne- follows from Eq. 47 that atoms prepared in a state with Þ ϭ glected, and the Coulomb term will only be included through m 0 will ionize much more slowly than atoms with m 0. a first-order Taylor expansion: 2  1/2 F. Ionization in a slowly varying field 2 F ͉p͉͑͒Ϸͫͩ ͪ Ϫ Ϫ ͬ 2 4 We now consider the situation when the external field is oscillating, 1/2  F 2 Ϸ ͫ1Ϫ ͬ Ϫ . ͑41͒ F͑t͒ϭF cos͑t͒zˆ. ͑48͒ 2 2 ͑2ϪF͒1/2 We assume the adiabatic approximation to be valid, that is, The integral in the exponent therefore contains two terms. the time the electron uses to tunnel through the barrier is The first term is much shorter than an optical period corresponding to the FЈ 1/2 3 regime with ␥Ӷ1 ͓see Eq. ͑2͔͒. Because the atom adiabati- ͵ ͩ Ϫ ͪ ЈϷϪ ϩ ͑ ͒ 1 2 d . 42 cally follows the changes in the external field, we can obtain 2 3F 2 0 the instantaneous ionization rate by inserting ͉F(t)͉ in the The second term is ͓see Ref. 11, Eq. ͑2.211͔͒ solution obtained with a static field. Because typical frequen-
253 Am. J. Phys., Vol. 72, No. 2, February 2004 C. Z. Bisgaard and L. B. Madsen 253 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 cies will correspond to wavelengths in the infrared region, reliability of our approximations. This question has been ad- the observed ionization rate will be an average of the instan- dressed both theoretically and experimentally, implying that taneous ionization rate over one period of the field. within the approximations outlined in this paper ͑␥Ӷ1 and We obtain the observed ionization rate from F/3Ӷ1), the model gives a good description of the ioniza- 1 /2 tion of atoms in strong fields. In fact these limits are too ϭ ͵ ͑ ͒ ͑ ͒ ͑ ͒ Wobs W t d t , 49 strict; the model is considered to be valid for ␥Ͻ0.5 and for Ϫ /2 fields weak enough that we can maintain the picture of a where W(t) is the instantaneous rate. Here the exponential barrier through which the electron tunnels. An example of dependence on the field strength clearly shows that the domi- the experimental tests of the tunneling theory can be found in nant contribution to the rate will be when ͉cos(t)͉ is near its Ref. 12. Note, however, that to compare the obtained rate maximum, that is, tϳ0. Because the exponential is much with experimental data, we must integrate the rate equations more rapidly changing than any power, the time dependence using the appropriate pulse profile both temporally and spa- of the preexponential factor will be neglected. For tϳ0, we tially. For a brief summary of the theoretical comparison make a Taylor expansion of 1/cos(t) and the time- between the static field ionization rates obtained from the dependent rate is approximated by ADK model with those obtained by directly solving the ¨ 3 Schrodinger equation, see Ref. 2. ͑ ͒Ϸ ͩ Ϫ ͑ ͒2 ͪ ͑ ͒ We have presented a coherent derivation of a much used W t Wstat exp t , 50 3F tunneling formula for strong field ionization of atoms. The ͑ ͒ derivation consists of six parts, Secs. II A–II F. Each of these where Wstat is the rate in a static field as in Eq. 47 . The observed rate is obtained as parts may be formulated as an exercise. We hope that the present note will be a useful supplement for teaching atomic, 1 /2 3 Ϸ ͵ ͩ Ϫ ͑ ͒2 ͪ ͑ ͒ molecular, and optical physics. Wobs Wstat exp t d t Ϫ/2 3F 3F 1/2 ϭͩ ͪ ͑ ͒ 3 Wstat , 51 where the integration limits were extended to infinity assum- ACKNOWLEDGMENT ing that 3/Fӷ1. For a slightly elliptical field L.B.M. was supported by the Danish Natural Science Re- search Council ͑Grant No. 51-00-0569͒. FϭF͓cos͑t͒xˆϮ sin͑t͒yˆ͔, ͑52͒ with Ӷ1, the result is a͒Electronic mail: [email protected] 1 ͑ ͒ ͑ ͒ 1/2 J. Giles, ‘‘The fast show,’’ Nature London 420,737 2002 . 3F 2 W Ϸͩ ͪ W , ͑53͒ Thomas Brabec and Ferenc Krausz, ‘‘Intense few-cycle laser fields: Fron- obs ͑1Ϫ2͒3 stat tiers of nonlinear optics,’’ Rev. Mod. Phys. 72, 545–591 ͑2000͒. 3P. Lambropoulos, P. Maragakis, and J. Zhang, ‘‘Two-electron atoms in 3 2 assuming that /Fӷ1Ϫ . strong fields,’’ Phys. Rep. 305, 203–293 ͑1998͒. When the light is circularly polarized, the magnitude of 4P. B. Corkum, ‘‘Plasma perspective on strong-field multiphoton ioniza- the electric field is constant. In this case the ionization rate tion,’’ Phys. Rev. Lett. 71, 1994–1997 ͑1993͒. will be equal to the static rate, and thereby be much larger 5B. Walker, B. Sheehy, L. F. DiMauro, P. Agostini, K. J. Schafer, and K. C. than the rate for linearly polarized light. Kulander, ‘‘Precision measurement of strong field double ionization of ͑ ͒ ͑ ͒ helium,’’ Phys. Rev. Lett. 73, 1227–1230 ͑1994͒. Equations 47 and 51 are the main results of the present 6 paper. As an application of these expressions, we suggest the L. D. Landau and E. M. Lifshitz, Quantum Mechanics: Non-Relativistic Theory ͑Butterworth-Heinemann, London, 1991͒, 3rd ed. calculation of the ionization rates for different intensities for 7B. M. Smirnov and M. I. Chibisov, ‘‘The breaking up of atomic particles Ar and Xe. In Ref. 10 the relevant values for the coefficient by an electric field and by electron collisions,’’ Sov. Phys. JETP 22, 585– D can be found by making the identification D 592 ͑1966͒. ϩ ϭ à ÃZ/ 1/2 à à 8 Cn l , where Cn l is the normalization constant A. M. Perelomov, V. S. Popov, and M. V. Terent’ev, ‘‘Ionization of atoms of the Coulomb wave function used in Ref. 10 with nà in an alternating electric field,’’ Sov. Phys. JETP 23, 924–934 ͑1966͒. ϭ à 9L. V. Keldysh, ‘‘Ionization in the field of a strong electromagnetic wave,’’ Z/ and l the effective principal and angular momentum ͑ ͒ ͑ ͒10 Sov. Phys. JETP 20, 1307–1314 1965 . quantum numbers. Ammosov, Delone, and Krainov ADK 10M. V. Ammosov, N. B. Delone, and V. P. Krainov, ‘‘Tunnel ionization of list experimental results that can be used for comparison with complex atoms and of atomic ions in an alternating electromagnetic field,’’ the theory. Sov. Phys. JETP 64, 1191–1194 ͑1986͒. 11I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products III. SUMMARY ͑Academic, New York, 1965͒. 12S. F. J. Larochelle, A. Talebpour, and S. L. Chin, ‘‘Coulomb effect in Now that we have obtained these relatively simple expres- multiphoton ionization of rare-gas atoms,’’ J. Phys. B 31, 1215–1224 sions for the ionization rate, it is worthwhile to consider the ͑1998͒.
254 Am. J. Phys., Vol. 72, No. 2, February 2004 C. Z. Bisgaard and L. B. Madsen 254 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