6 September 1999

Ž .Physics Letters A 260 1999 149–155www.elsevier.nlrlocaterphysleta

Perturbative study of multiphoton processes in the tunnelingregime

Marco Frasca 1

Via Erasmo Gattamelata, 3, 00176 Roma, Italy

Received 27 May 1999; received in revised form 6 July 1999; accepted 16 July 1999Communicated by B. Fricke

Abstract

A perturbative study of the Schrodinger equation in a strong electromagnetic field with dipole approximation is¨accomplished in the Kramers–Henneberger frame. A proof that just odd harmonics appear in the spectrum for a linearpolarized laser field is given, assuming that the atomic radius is much lesser than the free-electron quiver motion amplitude.Within this approximation a perturbation series is obtained in the Keldysh parameter giving a description of multiphotonprocesses in the tunneling regime. The theory is applied to the case of hydrogen-like atoms: The spectrum of higher orderharmonics and the above-threshold ionization rate are derived. The ionization rate computed in this way determines theamplitudes of the harmonics. The wave function of the atom proves to be rigid with respect to the perturbation so that theeffect of the laser field on the Coulomb potential in the computation of the probability amplitudes can be neglected as a firstapproximation: This approximation improves as the ratio between the amplitude of the quiver motion of the electron and theatom radius becomes larger. The semiclassical description currently adopted for harmonic generation is so rederived bysolving perturbatively the Schrodinger equation. q 1999 Elsevier Science B.V. All rights reserved.¨

Availability of powerful sources of laser light haspermitted, in recent years, the realization of experi-ments through gaseous media that have shown sev-eral new physical effects as photoionization with anumber of photons absorbed by the electron wellabove the ionization threshold and generation of a

w xbroad range of harmonics of the laser frequency 1 .This latter effect could have a lot of technologicalapplications and, as such, has been widely studiedboth theoretically and experimentally.

The possibility to turn a physical effect into apractical application is strongly linked with the avail-ability of a satisfactory theoretical model. But, it is

1 E-mail: marcofrasca@mclink.it

common belief that, due to the intensity of the laserfield, no perturbation theory can be done. The mainaim of this paper is then to show how perturbationtheory can be straightforwardly applied also for in-tense laser fields and analytical expressions can becomputed for any kind of multiphoton process, atleast for hydrogen-like atoms. The development pa-rameter turns out to be the the square root of theratio between the ionization energy I and the pon-B

deromotive energy U proportional to the intensityp

of the laser field, known in literature as the Keldyshparameter g . The regime of a small Keldysh parame-ter characterize the so-called tunnelling regime thatis the one of interest here.

Theoretical approaches to multiphoton processesare non-perturbative in nature and resort to Floquet

0375-9601r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved.Ž .PII: S0375-9601 99 00487-9

( )M. FrascarPhysics Letters A 260 1999 149–155150

w xtheory as in 2 , numerical methods applied directlyw xto the Schrodinger equation as done firstly in 3 or¨

w xsemiclassical models 4 . On the basis of the semi-classical ideas, a quantum theory for harmonic gen-eration has been obtained by L’Huillier and cowork-

w xers in 5 : Our theory permits to justify the mainassumptions of the quantum theory of these authors,so that, in turn, the semiclassical ideas prove to be afairly good description of harmonic generation.

The approach we apply to the Schrodinger equa-¨tion for an atom in an electromagnetic field can beeasily understood using a two-level model, widely

w xused for harmonic generation 6 . This model has theŽHamiltonian here and in the following we will take

."scs1

v0Hs s qVcos v t s , 1Ž . Ž .3 12

v being the level separation, V the intensity of the0

laser field and v its frequency, s and s are Pauli1 3

matrices. If V is small with respect to v , standard0

perturbation theory applies by interaction picturethrough an unitary transformation that removes theunperturbed part of the Hamiltonian: This gives aDyson series in the small development parameter

Ž .Vr v "v , out of resonance. Recently, duality has0w xbeen introduced in perturbation theory 7 and a dual

interaction picture has been devised where one doesan unitary transformation to remove the perturbation.For the above Hamiltonian one has to take Us

VŽ Ž ..exp yis sin v t that yields the transformed1 v 0

w xHamiltonian 8

vV0 Ž .2 is sin v t1 v0H s eF 2

v 2V v 2V0 0 i ns v t1s s J s q J e s ,Ý3 0 3 n 3ž / ž /2 v 2 vn/0

2Ž .

where now perturbation theory can be done forV4v ,v. We see straightforwardly that the unper-0

turbed part of the Hamiltonian is ‘dressed’ by thelaser field and so, the energy levels are shifted. Thenthe perturbation has odd and even harmonics of thelaser frequency and both can appear in the spectrum.But, probability amplitudes that enters in the compu-tation of the spectrum do not depend on the unitary

transformations one does on the Hamiltonian and thestates. So, we have sketched the physics of thetwo-level model in an intense monochromatic fieldjust through dual interaction picture. Although, as wewill show, the two-level model does not apply forcurrent experiments with atomic samples as in thiscase one observes just odd harmonics in agreementwith our full theory and it is not just a problem of aproper experimental setup, nevertheless it could havea wide range of applications in magnetic resonanceexperiments, for some other kind of media as optical

w xcavities 9 or wherever the conditions one meets foratomic samples are no more fulfilled.

The dual interaction picture applies in the sameway also to the Schrodinger equation in a semiclassi-¨cal laser field and in the dipole approximation, as

w xcurrently treated in literature 5 . The correspondencewith the two-level model above is remarkable. TheHamiltonian in this case is

p2 e e22Hs qV x q A t pq A t . 3Ž . Ž . Ž . Ž .

2m m 2m

By the unitary transformation

e e2t tX X X X2U t sexp yi dt A t py i dt A tŽ . Ž . Ž .H Hž /m 2m0 0

the above Hamiltonian transforms into

p2†H sU t qV x U tŽ . Ž . Ž .KH ž /2m

p2

s qV xqa t 4Ž . Ž .2m

e X XtŽ . Ž .with a t sy H dt A t . This is the well-known0m

Kramers–Henneberger Hamiltonian and the unitarytransformation above define the so called Kramers–

w xHenneberger frame 10 that shows as the effect ofthe electromagnetic field is to introduce a time-de-pendent translation on the potential of the unper-

Ž .turbed Hamiltonian by a length a t . The laser fieldcan be modeled as

E tŽ .A t sy xcos v t qj ysin v tŽ . Ž . Ž .ˆ ˆ

2(v 1qj

for a general ellipticity parameter j . Here we con-sider the simplest case of a linear polarization js0

( )M. FrascarPhysics Letters A 260 1999 149–155 151

Ž .and an instant rising of the laser field, that is E t sw xconst. So, one has 1

p2 dxX V xyxX , y , zŽ .lLH s qHKH X 222m pyl (L l yxL

q`kk i k v t yi k v tq i e q y1 e Õ x ,Ž . Ž .Ý k

ks1

5Ž .with

xX

TX k ž /dx ll LL XÕ x s V xyx , y , z , 6Ž . Ž . Ž .Hk X 22pyl (L l yxL

Ž . Ž Ž ..T x scos karccos x is the k-th Chebyshev poly-k

nomial of the first kind and l s eErmv 2L

s 4U rm 1rv the maximum free-electron quiver( p

motion excursion. This length is pivotal in the studyof atoms in an intense laser field as generally onehas l 4a, as1rmZe2 being the Bohr radius.L

One can see that, as for the two-level model, wehave the potential of the unperturbed part of theHamiltonian ‘dressed’ by the laser field and all theharmonics, odd and even, are present in the perturba-tion. We can now show that, in all the current

Ž .experiments where the potential V x depends just< <on rs x and l 4a, a being the Bohr radius ofL

the atoms in the sample, then just odd harmonicsappear in the spectrum. Indeed, we can rewrite Eq.Ž .6 as

1 2X X 2 2(Õ x s dx V xyl x qy qzŽ . Ž .Hk Lž /y1

=T xXŽ .k

. 7Ž .X 2(p 1yx

If the laser field is enough intense, a series in arlLŽ .is obtained if one develops Eq. 7 in Taylor series as

Õ xŽ .k

1 2X X 2 2(s dx V xyl x qy qzŽ .H Lž / xs0, ys0, zs0y1

=T xX xXŽ . 1k X X X< <yx dx V l xŽ .H XLX 2 < <x( y1p 1yx

=T xXŽ .k

q . . . . 8Ž .X 2(p 1yx

Despite its appearance, the terms of this series can beevaluated for a Coulomb potential and proved to befinite assuring the convergence. This is due to thefact that in this case the integrals can be computedanalytically. Then from the above expression twomain conclusions can be drawn. Firstly, multiphotoneffects are due to a dipole induced on the atom in thesame direction as the electric field of the laser andsecondly, Chebyshev polynomials have a definiteparity and due to the symmetrical range of integra-tion, only odd polynomials give a non-null contribu-tion to the second term, while the first term has nophysical consequences and in the following will beneglected. So, only odd harmonics contribute to thespectrum while, even harmonics are quadrupole radi-ation and then strongly depressed. Indeed, for aCoulomb potential one obtains

x Ze2n

Õ x fyi y1 2nq1 . 9Ž . Ž . Ž . Ž .2 nq1l lL L

This result, that does not involve any other approxi-mation beside the symmetry of the potential and theamplitude of the quiver motion of the electron withrespect to the atomic radius, supports in some way

w xthe physical view recently given in 11 , where it isassumed that the electron recolliding with the atomiccore, emits bremsstrahlung radiation that is cut off atthe maximum amplitude of the quiver motion of theelectron, producing in this way just odd harmonics.

To complete the above discussion before introduc-ing perturbation theory, we have to study the‘dressed’ potential Õ . This should be managed dif-0

ferently from the time-dependent part. Indeed, weŽ .have to separate the original potential V r from the

shifts induced by the laser field on the energy levelsof the atom. This can be obtained by a Taylorexpansion as

2X 2 2X (V xyl x qy qzŽ .dx Lž /1Õ x sŽ . H0 X 2p (y1 1yx

sV r qd V xŽ . Ž .L

2lL X X2 2sV r q V r y qV r zŽ . Ž . Ž .34rXX 2qV r x r q . . . , 10Ž . Ž .

( )M. FrascarPhysics Letters A 260 1999 149–155152

where is seen that only even terms survive andhigher order terms fall off very rapidly with r. Theabove expression assumes a very simple form for aCoulomb potential

2 n2 q`Ze l xLÕ x sy 1q A P ,Ž . Ý0 n 2 n ž /ž /r r rns1

11Ž .1 2 n 2'Ž .A sH dxx r p 1yx and P is the n-th Leg-n y1 n

endre polynomial. This way to express the dressedCoulomb potential gives us a way to prove that thewave function is ‘rigid’ with respect to the perturba-tion using standard Rayleigh–Schrodinger perturba-¨tion scheme, for the kind of problems we discusshere. But, it should be pointed out that for stabiliza-

w xtion things are quite different 12 .The equations for the amplitudes are given by

˜ ˜yi ŽE yE . tn m² < < :ia t s a t m d V x n eŽ . Ž . Ž .˙ Ým n Ln/m

q`

k ² < < :q i a t m Õ x nŽ . Ž .Ý Ý n kn ks1

=˜ ˜yi ŽE yE yk v . tn me

k ˜ ˜yi ŽE yE qk v . tn mq y1 e 12Ž . Ž .˜ ² < Ž . < : Ž .having set E sE q n d V x n , d V x beingn n L L

the part of the static potential due to the laser field.At this point, all the machinery of standard perturba-

w xtion theory applies 13 . For our aim, we have toshow that the Rayleigh–Schrodinger part gives in-¨deed a small contribution to the amplitudes. Byassuming the atom initially in its ground state, thiscontribution is

² < < :m d V x 1Ž .L ˜ ˜RS yiŽE yE . t1 ma t f e y1 . 13Ž . Ž .Ž .m ˜ ˜E yE1 m

Ž .Using Eq. 11 is easy to verify that no contribution² < Ž . < :comes for ms2 as 2 d V x 1 s0 but the degen-L

eracy of level 2 is removed by the dressed potential² < Ž . <as one has ms2,ls1,l s0 d V x ms2,lsz L

: 2 Ž .Ž .2 ²1,l s0 sZe r 240a l ra and ms2,ls1,z L< Ž . < : 2l s " 1 d V x m s 2,ls1,l s"1 syZe rz L z

Ž .Ž .2 ² < Ž . <480a l ra while ms2,ls0,l s0 d V x mL z L:s2,ls0,l s0 s0. Indeed, one can see that allz

the states having m even do not give a first ordercontribution even if the level shift is not null, whilethe level-shift is always 0 when ls0. Instead, for

² < Ž . <ms3 one has e.g. ms3,ls2,l s0 d V x msz L2 2': Ž .Ž .1,l s 0,l s 0 s Ze 150 r 10800a l ra andz L

² < Ž . <for the level shifts ms3,ls2,l s0 d V x msz L: 2 Ž .Ž .2 23,l s 2,l s 0 s Ze r 5670a l ra yZe rz L

Ž .Ž .4 ² < Ž . < :136080a l ra and 1 d V x 1 s0, so theL LŽ .correction of Eq. 13 turns out to be

2'150 lLž /10800 aRSa t fyŽ .3,2,0 2 44 1 l 1 lL Lq yž / ž /9 5760 a 136080 a

=8

i E t1e y1 , 14Ž .9ž /which is indeed negligible and the wave functionturns out to be ‘rigid’ with respect to the deforma-tions introduced by the laser field. This is even moretrue as larger become the ratio l ra. The reason forL

Ž .this is that only a finite number of terms of Eq. 11give a non-null contribution to the matrix elements.It is interesting to note that for stabilization of anatom in intense laser field the situation is exactly thecontrary as one should be able to diagonalize the

2 Ž .Hamiltonian H sp r2mqÕ x being the time-0 0

dependent part negligible, an approximation that be-comes exact in the limit of infinite frequency of the

w xlaser field 1,12 .Ž .Then the iterative procedure to solve Eq. 12 can

be applied to compute the probability transition forany process. This approach implies that off-resonantcontributions should be systematically neglected. Inthis way, a golden rule is straightforwardly obtainedas

q`2 ˜ ˜<² < < : <P s2p i Õ x f d E yE ynv ,Ž .Ýi™ f n f i

ns1

15Ž .

from which several results for multiphoton processescan be obtained. It is assumed a continuum of finalstates to sum over so that excited levels can decay,

w xotherwise quantum resonance theory applies 15 andRabi flopping is obtained. In any case, going tosecond order gives a.c. Stark shifts of the energylevels. Rabi frequency due to resonance with the k-thharmonic of the perturbation with two levels m and

Ž w x. <² < Ž . < : <n of the atom is Ref. 15 V r2s m Õ x n .R k

( )M. FrascarPhysics Letters A 260 1999 149–155 153

Ž .From Eq. 15 we can easily compute the rate ofabove threshold ionization. For hydrogen-like atomsw Ž .xEq. 9 and assuming the atom initially in its groundstate one has

5r22 q`32 v IB2Gs g Ý3 U 2nq1 vŽ .p nsn0

=

3r2IB1y , 16Ž .

2nq1 vŽ .Ž .n being the minimum integer for which 2n q1 v0 0

y I G0. It has been used the fact that, as shownB

above, for the ground state of hydrogen-like atomsthere is no shift by the part of the static potential due

² < Ž . < :to the laser field, that is 1 d V x 1 s0 forL

Coulomb potential. Beside, a plane wave is assumedfor the particle in the final state to make computation

w xsimpler. By taking Ref. 14 for experimental results,we can check the above expression for helium andneon that show a large plateau in the tunnelingregime. So, we have U s 155 eV being the inten-p

sity 1.5 = 1015 Wrcm2, vs 1.177 eV and I sB

24.59 eV. Then gf .4 and Gf 0.026 eV, that issmall as it should be expected. The same computa-tion for neon gives approximatively 0.02 eV.

To analyse the question of harmonic generation,² : ² Ž . < < Ž .:one has to compute x s C t x C t . To com-

plete this computation, we assume that no intermedi-ate resonance is present and will justify this assump-tion a posteriori through the quantum resonance the-

w xory of Ref. 15 , that here applies. So, let us take anatom initially prepared in its ground state as to haveŽ . Ž .a 0 sd . From Eq. 12 one hasi i0

² < < :m d V x 0Ž .L ˜ ˜yi ŽE yE yie . t0 ma t sd q eŽ .m m0 ˜ ˜E yE y ie0 m

q`

k ² < < :q i a t m Õ x 0Ž . Ž .Ý n kks1

=

˜ ˜yi E yE yk vyie tŽ .0 me

˜ ˜E yE ykvy ie0 m

˜ ˜yi E yE qk vyie tŽ .0 mekq y1 q . . . ,Ž .˜ ˜E yE qkvy ie0 m

17Ž .

1with the limit e™0 understood as to have sx" i01 Ž .P . ipd x , P being the principal value. As isx

customary in perturbation theory, we keep just thoseterms that are near resonant with the harmonics ofthe perturbation: The only possibility left is thecontinuous spectrum, as it should be with the currentunderstanding of harmonic generation. So, we take

q`kk² < < :a t fy i p Õ x 0 y1Ž . Ž . Ž .Ýp k

ks1

=

˜iŽE yE yk vqie . tp 0e, 18Ž .˜E yE ykvq iep 0

p being the momentum of the particle in the continu-ous part of the spectrum. Now, we specialise thisexpression to the case of hydrogen-like atoms having

2 q`Ze² < < :a t f 2nq1 p x 0Ž . Ž .Ýp 2lL ns0

=eiŽEpyE 0yŽ2 nq1.vqie . t

. 19Ž .E yE y 2nq1 vq ieŽ .p 0

Then for the dipole moment one has

² : yi ŽEpyE 0 . t² < < :x f a t e 0 x p qc.c. 20Ž . Ž .Ý pp

After passing from the sum to integration through3 Ž .3Ý ™VHd pr 2p and taking for the final state ap

plane wave, one gets the final expression for theharmonic spectrum

2 q` 3r264 Ze v xn5² :x fy g Ý9r2 2 523 Up gnsn0x qnž /3

=sin 2nq1 v t , 21Ž . Ž .Ž .Ž .where x s 2nq1 vy I r3U . The normalizationn B p

to 3U for x originates from the fact that, from thep n

above expression, the intensities of the harmonicsreduce as the factor 3U increases. Then if thep

Keldysh parameter g is enough small we can take

2 q`64 Ze v 15² :x fy g sin 2nq1 v t ,Ž .Ž .Ý9r2 2 7r23 U xp nnsn0

22Ž .

( )M. FrascarPhysics Letters A 260 1999 149–155154

so that only for x F1 the harmonic amplitudes aren

large. This is the approximate cut-off law found outw xthrough semiclassical methods in Ref. 4 . It should

also be stressed the existence of a minimum har-monic order n that should be expected due to the0

close connection between harmonic generation andmultiphoton ionization. Indeed, this lower boundcomes out from the phase space through the integra-tion of the Dirac function both for the golden ruleŽ .15 and for the computation of the dipole moment² :x . Then one gets n s10 and 9 for helium and0

neon respectively, that means harmonic 21 for thestarting point of the spectrum in the regime ofinterest. It should be pointed out that the above

² :equation for x has to take properly into accountŽ .the ionization rate G of Eq. 16 as to have at last

2 q` 3r264 Ze v xn5² :x fy g Ý9r2 2 523 Up gnsn0x qnž /3

=sin 2nq1 v t eyG t . 23Ž . Ž .Ž .

One can estimate the constant factor that determinesthe amplitude of the harmonics 64r39r2 Ze2vrU 2g 5.p

Indeed, for helium one obtains approximately 0.32= 10y8 eVy1 and for neon about 0.12 = 10y7

eVy1, showing, as it should be, a larger amplitudefor neon.

A further analysis concerns the effect of interme-diate resonances on the spectrum of harmonics. On

w xthe basis of the theory of Ref. 15 , one can writeŽ .down Eq. 18 as

q`kk² < < :a t fy i p Õ r 0 y1Ž . Ž . Ž .Ýp k

ks1

=

˜iŽE yE yk vqie . tp 0e VRcos t , 24Ž .ž /˜ 2E yE ykvq iep 0

with V the Rabi frequency computed taking inR

account the resonances between the ground state andother discrete levels. To compute the above expres-sion we assumed that the atom is initially prepared in

VRŽ .its ground state so that, a t scos t , essentiallyŽ .0 2

the rotating wave approximation. It is easy to realize

that one gets the harmonics in the spectrum shiftedVRby the quantity " .2

The theory above could have wide applicabilityas, in principle for any multiphoton process one isable to compute analytical formulae to compare withexperimental results. For instance, an improvementeasy to implement is to use a full Coulomb wavefunction also for the final state in the above compu-tations. On the other hand, even if major features ofmultiphoton processes are described by this theory,several problems are surely opened up as the appli-cability of the theory for an ellipticity parameterj/0, the introduction of a slower rising of the laserfield or how to take into account all the features thatreal experiments have for harmonic generation. Be-side, when the intensity of the laser field becomestoo high the above approach should be properlymodified as relativistic effects enter into the physicalpicture and, e.g. even harmonics can also be signifi-

w xcant 16 . Experiments to generate even harmonicsw xare also carried out using solid surfaces as in 17 .

Anyhow, it should be stressed how the possibility toderive a perturbative solution to the Schrodinger¨equation could give a chance to check models ofmultiphoton physics that no other approach offers.

References

w x1 M. Protopapas, C.H. Keitel, P.L. Knight, Rep. Prog. Phys. 60Ž .1997 389.

w x Ž .2 R.M. Potvliege, R. Shakeshaft, Phys. Rev. A 40 1989 3061.w x Ž .3 K.C. Kulander, B.W. Shore, Phys. Rev. Lett. 62 1989 524;

Ž .J.H. Eberly, Q. Su, J. Javanainen, Phys. Rev. Lett. 62 1989881.

w x4 K.C. Kulander, K.J. Schafer, J.L. Krause, Phys. Rev. Lett. 68Ž . Ž .1992 3535; P.B. Corkum, Phys. Rev. Lett. 71 1993 1994.

w x5 M. Lewenstein, Ph. Balcou, M.Yu. Ivanov, A. L’Huillier,Ž .P.B. Corkum, Phys. Rev. A 49 1994 2117.

w x6 A first indication that this model gives a nice description ofharmonic generation was yielded in B. Sundaram, P.W.

Ž .Milonni, Phys. Rev. A 41 1990 6571. Recent applicationsare, e.g., F.I. Gauthey, C.H. Keitel, P.L. Knight, A. Maquet,

Ž .Phys. Rev. A 52 1995 525; M.L. Pons, R. Taieb, A.Ž .Maquet, Phys. Rev. A 54 1996 3634.

w x Ž .7 M. Frasca, Phys. Rev. A 58 1998 3439.w x Ž .8 M. Frasca, Phys. Rev. A 56 1997 1548.w x Ž .9 P. Meystre, Opt. Commun. 90 1992 41; M. Wilkens, P.

Ž .Meystre, Opt. Commun. 94 1992 66.

( )M. FrascarPhysics Letters A 260 1999 149–155 155

w x10 H.A. Kramers, Collected Scientific Papers, North-Holland,Amsterdam, 1956; W.C. Henneberger, Phys. Rev. Lett. 21Ž .1968 838.

w x11 M. Protopapas, D.G. Lappas, C.H. Keitel, P.L. Knight, Phys.Ž .Rev. A 53 1996 R2933.

w x12 M.V. Fedorov, Atomic and Free Electrons in a Strong LightField, World Scientific, Singapore, 1997

w x13 A. Messiah, Quantum Mechanics, North-Holland, Amster-dam, 1961, Vol. II, Ch. XVII.

w x Ž .14 A. L’Huillier, Ph. Balcou, Phys. Rev. Lett. 70 1993 774.w x Ž .15 M. Frasca, Phys. Rev. A 58 1998 771.w x Ž .16 C.H. Keitel, P.L. Knight, Phys. Rev. A 51 1995 1420; C.H.

Ž .Keitel et al., J. Phys. B 31 1998 L75.w x Ž .17 D. von der Linde et al., Phys. Rev. A 52 1995 R25.