ISSN 1054�660X, Laser Physics, 2009, Vol. 19, No. 8, pp. 1540–1543.© Pleiades Publishing, Ltd., 2009.Original Russian Text © Astro, Ltd., 2009.

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Tunneling is an important quantum�mechanicaleffect that was used to interpret the α decay and pro�vided the triumph of quantum mechanics at earlystages.

We can assume that the theoretical analysis of tun�neling effects in atomic phenomena was started in [1],where the solution to the problem of the tunnelingionization of atomic hydrogen in the presence of astatic electric field was proposed. The analysis was per�formed on the assumption that, for the electronmotion in the presence of the Coulomb field, the sep�aration of variables can be performed in both sphericaland parabolic coordinates. The corresponding resultsare presented in many textbooks on quantummechanics (see, for example, [2]).

The next significant theoretical result was obtainedin [3], where the probability of the tunneling ioniza�tion of an arbitrary atom in the excited (Rydberg) statewas calculated. For an arbitrary atom, the spatial vari�ables that describe the motion of an excited electronare not separated in parabolic coordinates, whichleads to a substantial complication of the problem incomparison with the problem of the simplest (hydro�gen) atom. In addition, the arbitrariness of the atommust be taken into account in the theory. It was dem�onstrated in [3] that it is suffice to know only the quan�tum defect of the electronic state prior to the tunnelingionization for the calculation of the tunneling proba�bility. Recall that the quantum defect determines thedifference between the energy of the one�electronexcitation in an arbitrary atom and the energy of a sim�ilar state in the hydrogen atom.

The classical works on quantum mechanics weresupplemented with the famous work [4], where thepossibility of tunneling in the presence of an ac fieldwas demonstrated under the condition for a relativelyhigh field strength F and a relatively small field fre�

quency ω. Parameter γ, known as the Keldysh param�eter, must satisfy the following condition:

(1)

where m is electron mass and E in the bond energy inatom. When the opposite inequality is satisfied, a con�ventional (one�photon) atomic ionization takes placeprovided that the energy of the electromagnetic pho�ton is sufficient for bond breaking or the multiphotonionization occurs when the energy of one photon isinsufficient. Thus, the dependence of the probabilityof the photoeffect on the electric field strength for then�photon ionization is proportional to F2n and theprobability of the tunneling ionization is proportionalto exp(–F0/F), where F0 is the characteristic field.

These results were obtained in [4] for an electron ina short�range potential rather than an atom. In a cer�tain approximation, this potential is the potential forthe electron motion in a negative ion. Nevertheless, itwas demonstrated later that the importance of theKeldysh parameter for the classification of the photo�effect is retained for alternative objects (e.g., neutralatoms and molecules and their positive ions).

Naturally, the Keldysh work has been stimulated bythe interest in new laser sources as the instruments fornew physical observations. The same is valid for theexperiments on the multiphoton ionization. The pres�ence of laser fields led to violations of the fundamentallaw of the photoeffect lying in the existence of thephotoelectric threshold, which was experimentallydemonstrated by Lebedev and was theoretically inter�preted by Einstein with the aid of the concept of theelectromagnetic energy quantization. Indeed, in thepresence of a sufficiently strong laser field, the photo�electrons can be generated owing to the absorption ofseveral photons although the photon energy is lessthan the bond energy.

The Keldysh work has stimulated the experimentalstudy of the atomic ionization by the laser field. From

γω

eF����� 2mE � 1,=

TUNNELING

Theory of Tunneling Ionization in the Presence of the Laser Field: History and State of the Art

B. A. ZonVoronezh State University, Universitetskaya pl. 1, Voronezh, 394006 Russia

e�mail: zon@niif.vsu.ru; zfira@yandex.ruReceived February 26, 2009

Abstract—The main physical concepts of the theory of the atomic tunneling ionization in the presence of astrong laser field and the supporting experimental evidence are presented.

PACS numbers: 32.80.Fb, 32.80.Wr

DOI: 10.1134/S1054660X09160099

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the experimental point of view, the simplest scenarioinvolves the application of noble gases as atomic tar�gets. Thus, the first experiments were performed withnoble gases. However, the observed photoelectronscould not be assigned to the noble�gas atoms. In fact,the photoelectrons were predominantly generated dueto the ionization of the organic molecules of oils in thevacuum chambers, since the ionization potential ofsuch molecules is significantly less than the ionizationpotential of the noble�gas atoms.

For the first time, the problem was solved in [5],where Delone and Voronov proposed and imple�mented the mass�spectrometric observation of thepositive ions resulting from the photoionization ratherthan electrons. We can hardly overestimate the impor�tance of this work, since it has initiated the experimen�tal study of the multiphoton processes that are cur�rently investigated in the leading scientific laborato�ries.

Note that the multiphoton ionization rather thanthe tunneling ionization was considered in [5] and thesubsequent works devoted to various aspects of thephenomenon. My colleagues from the Voronezh StateUniversity and I took part in the theoretical interpre�tation of the experimental dependences after the All�Union Conference on Coherent and NonlinearOptics, which took place in Kiev in 1968. Since thattime, I could regularly discuss the problems of themultiphoton ionization with Delone and his cowork�ers. One of the main topics for the discussions was thepossibility of the experimental implementation of thecondition for the tunneling effect (expression (1)). Wedid not know that the strong inequality in condition(1) could be changed by the normal inequality, so thatthe power dependence of the probability of the photo�effect could be changed by the exponential depen�dence. The ruby and neodymium lasers with a photonenergy of about 1 eV were available in the Delone’slaboratory. The application of the CO2 laser with aphoton energy of about 0.1 eV immediately allowedthe observation of the tunneling ionization. Theexperiments were performed in [6], and the authorsexperimentally discovered the tunneling ionization inthe presence of an ac field.

Consider the theory of the tunneling effect. As wasmentioned, the results from [4] were obtained for aparticle in a short�range potential rather than an atom.The extension of the analysis to real atoms can befound in [7], where the authors employed theSmirnov–Chibisov formula [3] and replaced the staticelectric field by the harmonic (time�dependent) field.The validity of such an assumption in the framework ofthe adiabatic approximation is based on the fact thatthe field frequency is assumed small in the tunnelingregime. The averaging over the field period yields theformula for the tunneling ionization of an atom in thepresence of an ac laser field. The comparison of thetheoretical results with the experimental data from [6]can be found in [8], where an excellent agreement

between the theoretical and experimental data wasdemonstrated. This theory is known as the ADK the�ory (the first letters of the names of the authors of [8]).

In the modern experiments on the action of thestrong and superstrong laser radiation on matter, thetunneling ionization dominates and the multiphotonregime is considered as an exotic regime in contrast tothe first works on the multiphoton processes. This cir�cumstance is due to the application of ultrashort laserpulses when condition (1) is satisfied at all of the fre�quencies owing to a relatively high electric fieldstrength.

Note the new unexpected natural phenomena thatwere not theoretically predicted. Significant efforts ofexperimenters and theoreticians were needed for thecorrect interpretation of the effects. First, consider theformation of multicharged atomic ions. For the firsttime, the doubly charged ions were observed in themultiphoton regime for the divalent atoms (Ba, Sr,etc.) at the Uzhgorod University [9]. Unexpectedly,the probability of the formation of the doubly chargedions was comparable with the probability of the forma�tion of the singly charged ions although the theory pre�dicted a difference of several orders of magnitude. Inspite of the absence of the accurate calculations of theabove probabilities, we can assume that the divalentatoms are poorly described using the original Keldyshmodel based on the short�range potential. Apparently,the absorption of laser photons by a divalent atominvolves both valence electrons and the two�electronexcited states are virtually populated.

However, in the presence of stronger fields with anintensity of up to 1016 W/cm2, under the conditions forthe tunneling regime (expression (1)), the formationof the multicharged ions results from the impact ion�ization by the fast electron (that is detached from theatom and, then, is accelerated in the laser field) ratherthan the direct action of the laser radiation. Such a res�cattering process was proposed in [10, 11] and wasexperimentally validated. In the presence of strongerfields with an intensity of greater than 1016 W/cm2, therescattering is suppressed, since the free electronmotion is affected by the magnetic component of theelectromagnetic field of the light wave. Thus, the elec�tron motion ceases to be rectilinear and the electrontrajectory becomes similar to a figure of eight, so thatthe electron does not arrive at the atom upon the light�induced oscillations. The corresponding experimentaldata can be found in [12]. Hence, the direct laser–atom interaction that leads to the formation of multi�charged ions plays a significant role at such high inten�sities (superstrong laser fields).

Evidently, the single�particle theory of the tunnel�ing effect (the ADK theory) can be inaccurate in theanalysis of the formation of multicharged ions. How�ever, one can hardly predict the relative importance ofthe multiparticle effects. The contributions of severaleffects must be analyzed, and the general physical sce�

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ZON

nario must be interpreted. The theoretical analysis isavailable for the simplest multiparticle effects, and wewill consider the corresponding experimental data.

First, note the inelastic tunneling effect, in whichthe ion core can be found in the excited state ratherthan the ground state (as in the case of the conven�tional tunneling). Obviously, this effect can beneglected when only the formation of the singlycharged ion is considered. However, at a sufficientlyhigh laser intensity, the resulting ion can be repeatedlyionized and the tunneling with the core excitation cansubstantially contribute to the total probability of theformation of the multicharged ion. The calculation ofthe probability of the inelastic tunneling and the anal�ysis of its role in the formation of the multichargedions can be found in [13]. The calculations employ theapproach from [14], where the one�photon two�elec�tron ionization of He was investigated. The collision ofthe He atom with a high�energy photon leads to thefast ionization of one electron. In the presence of thenew self�consistent field, the second electron canremain in the bound state of the He+ ion or can be ion�ized owing to the nonorthogonality of the electronwave functions in the neutral He and He+. Such a pro�cess of the formation of He2+ is similar to the knownshock approximation [2] but the difference lies in thefact that the energy expended by the ionization of thesecond (slow) electron is taken into account in the cal�culation of the ionization probability for the first (fast)electron. The same specified shock approximation wasused in the calculation of the inelastic tunneling in[13]. A variation in the self�consistent field in theexternal shell was disregarded, and, hence, the excitedstates of the residual ion represent alternative compo�nents of the fine structure of the multiplet to which theground state of the residual ion also belongs.

The probabilities of the formation of the Ar and Krmulticharged ions were calculated in [15] with allow�ance for the inelastic tunneling. Moderate laser fieldswere considered, and the circularly polarized laserradiation was used to eliminate the rescattering. Thetheoretical results for Ar were experimentally provenin [16]. In addition, the probability of the Ne2+ ioncreation calculated in [17] is in good agreement withthe experimental data from [18] obtained for the cir�cularly polarized laser field. Thus, the theory of theinelastic tunneling effect is experimentally validated.

The analysis of an atom as a multiparticle systemmakes it possible to interpret both the inelastic tunnel�ing and the fast relaxation of electrons of the ion corewith respect to the magnetic quantum number (mrelaxation), which prepares the electrons for tunnel�ing. Note that the maximum tunneling probability isreached when the projection of the electron orbitalmomentum along the direction of the electric field(magnetic quantum number m) is zero. When the firsttwo electrons from the state m = 0 are detached, theremaining electrons have |m | > 0 in the framework of

the single�particle model. A possibility of the slowingof the tunneling upon the formation of ions with adegree of ionization of greater than 3 was mentionedin [19]. However, the corresponding experiment from[20] did not prove the assumption: the tunneling prob�ability did not decrease. The interpretation of the fastm relaxation involves the interelectron interaction inthe ion core. The corresponding relaxation time wasestimated in [21] using the energy–time uncertaintyrelation: under typical conditions, the relaxation time(5 × 10–16 s) is significantly less than the laser pulseduration.

Another multiparticle effect known as the collec�tive tunneling lies in the simultaneous detachment ofseveral electrons from an atom. The correspondingcalculations can be found in [22]. However, the effectwas insignificant in the calculations of the formationof the Ar and Kr multicharged ions [15]. The collectiveeffect becomes significant when the probabilityincreases by no less than an order of magnitude in thepresence of stronger fields with intensities of greaterthan 1019 W/cm2 [21].

The theory was developed to take into account boththe multiparticle effects and the relativistic effects thatare important for the multicharged ions in which theelectron bond energy at the outer shells is comparablewith the electron rest energy. The corresponding anal�ysis can be found in [23, 24]. Note also the well�knownwork [25] in which the ADK theory is used for thestudy of molecules (mol�ADK).

ACKNOWLEDGMENTS

I am grateful to A.S. Kornev for the preparation ofthe article. This work was supported by the RussianFoundation for Basic Research (project no. 08�02�00337). I strongly hope that this work substantiallybased on personal memories will contribute to thememorial of Nikolai Borisovich Delone and his role inthe creation and development of the physics of mul�tiphoton processes.

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