Multiphoton absorption by molecules1

Pedro F. Gonza´lez-Dıaz

Centro de Fı´sica Miguel Catala´n, Instituto de Matema´ticas y Fısica Fundamental, Consejo Superior de Investigaciones Cientı´ficas, Serrano121, 28006 Madrid, Spain

Received 1 July 1996; accepted 15 February 1997

Abstract

Rabi frequencies for multiphoton absorption by atoms or molecules that can be characterized as two- or three-level systemsare obtained in the cases of single and double laser perturbations. If the ambiguity in the origin of absolute energy isconveniently gauged off in the three-level model, there appears an additional non-linear frequency which originates fromthe separate evolution of a relative frequency component induced in the whole system by the simultaneous action of the twoperturbations.q 1998 Elsevier Science B.V. All rights reserved

Keywords:Absorption; Multiphoton; Rabi frequencies

I first met Professor Ibo Smeyers in 1972 when Iwas starting my PhD work. Since then his leadingmagistry has continuously influenced my scientificwork, not just on subjects related to molecular physicsand theoretical chemistry, but also on others apparentlyunrelated to his main fields of interest. It is for me both apleasure and an honor to contribute this volume com-memorating the occasion of his 65th anniversary witha work that can be singled out as an example of theinfluence of I. Smeyers in my scientific endeavor.

Since multiphoton absorption and dissociation insmall polyatomic molecules were discovered twodecades ago [1,2] there has been a considerableinteres [3,4] in the study of the mechanism by whicha single monochromatic infrared laser radiation can beabsorbed by a highly anharminic molecular mode.This interest has led to a variety of models whichrange from those suggesting a fully stochastic

interpretation [5] to models invoking rotovibrationallyassisted contributions [6,7]. Among these models, thereemerged the quite interesting possibility that intra-molecular generation of deterministic chaos maybecome the main mechanism responsible for mono-chromatic multiphoton excitation of small molecules[8], with the Rabi frequency playing the role of thecontrol parameter. Taking the Rabi frequency as a con-stant parameter could be not quite correct an assump-tion as it becomes non-linerly dependent on radiationintensity at high intensities of the driving field [9] andtherefore no longer a genuine constant of motion.

This work aims at obtaining expressions for theRabi frequency in multiphoton processes using afully quantized procedure when the excited moleculescan be modelized as two- or three-level systems. Inaccordance with previous indications, it is obtainedthat such expressions generally depend non-linearlyon field intensity, so rendering the Rabi frequencyunsuitable as constant of motion. We use two simplemodels. We consider first a two-level system which

Journal of Molecular Structure (Theochem) 433 (1998) 59–62

THEOCH 5475

0166-1280/98/$19.00q 1998 Elsevier Science B.V. All rights reserved.PII S0166-1280(98)00011-6

1 Dedicated to Professor Yves G. Smeyers Guillemin on theoccasion of his 65th birthday.

coherently absorbsM monochromatic photons. Theresults from this model are compared with thoseobtained by other authors, and used to construct themore interesting three-level model where twoindependent radiation beams with different wave-lengths are considered. From this we derive themain result of this work, which is the appearance ofan additional non-linear absorption frequency that canbe described in terms of the relative frequency of thetwo driving fields. It will be seen that it is theexistence of this additional frequency thatdistinguishes a three-level system from a two-levelsystem with respect to multiphoton absorption.

Let us write the fully quantized Hamiltonian for theabsorption ofM photons with frequencyq by the two-level system with transition frequencyq0 in therotating wave approximation

H2 =HR +HM +HRM

="qa†a+ 12"q0jz +"b(a†Mj_−aMj + ) (1)

wherea, a†, the annihilation and creation operators forthe driving field, satisfy the usual commutationrelations [a,a†] = 1, andj is the 2× 2 Pauli matrixwhich, in turn, obeys the commutation relations:

[jz,j6] = 6 j6, [j + ,j − ] = jz

{j + ,j − } = I

with I the unit matrix;b is the dipole moment matrixelement for the simultaneous absorption ofMmonochromatic photons by the system.

The Heisenberg equations of motion read

jz =2ib(a†Mj − +aMj + )

j + = i(q0j + −ba†Mjz) (2)

a= − i(bMa†M −1j − +qa)

Introducing the constant of motion ([N,H] = 0)

N =a†a+Mj + j − (3)

we finally obtain

jz = −2(Mq −q0)

"H2 −"q(N − 1

2M)� �

− [(Mq −q0)2 −2lbl2MA2(n)]jz + [2lbl2MB2(n)]j2z

(4)

wheren = a†a and

A2(n) =n!

(n−M)!+

(n+M)!n!

(5)

B2(n) = ∑M −1

a =0

(n+a)!(n−M +a +1)!

(6)

If we take the Rabi frequency as the coefficient of thespin operatorjz, there could be some ambiguity insuch a definition using Eq. (4) because the last termin this expression, which depends onj2

z, could eitherbe absorbed into the definition of a unique Rabi fre-quency given by the coefficient ofjz (notice that theinsertion ofjz = 2j+j− − 1 in the last term of Eq. (4)allows one to factorize a given coefficient forjz

depending onj+j− in that term, and then by usingEq. (3), one may rearrange that coefficient, togetherwith the similar one in the second term, in such a waythat the resulting overall coefficient forjz will notexplicitely depend on operatorsj+j− and jz, but onN andn), or define by itself a new non-linear funda-mental frequency if we would interpret this as thecoefficient of the spin operator squaredj2

z. For thecase of a two-level system only the first possibilityapplies because, once the physically irrelevantenergy-origin gauge is subtracted, we are left withjust one single periodic evolution for the two eigen-states in the semiclassical treatment; i.e. for two-levelsystems, one can always have a one-body representa-tion of the pure molecular Hamiltonian and thisshould associate with a single Rabi frequency givenby

Q2R = (Mq −q0)2 +2lbl2[MA2(n) −B2(n)(M −2N +2n)]

(7)

Thus, QR is no longer a constant of motion anddepends explicitly onn non-linearly forM . 1. Simi-lar expressions have been obtained by Sukumar andBuck [10] and Kochetov [11]. We note that Eq. (7)reduces to the known Knight–Milonni equation [12]for the case of linear absorptionM = 1.

The existence of only a single linear Rabifrequency,QR, is no longer a characteristic of multi-photon absorption by anN-level system whenN . 2.In dealing with the molecular Hamiltonian as a levelsystem in terms of the absolute energies of the levels,one always has an ambiguity in the choice of theorigin for such energies. This ambiguity is eliminated

60 P.F. Gonza´lez-Dıaz/Journal of Molecular Structure (Theochem) 433 (1998) 59–62

by re-expressing the Hamiltonian in terms of the leveltransition energies rather than absolute energies. Inthe case of a three-level system,N = 3, this can bedone by writingHM as (Fig. 1):

HM =12{ (E1 +E3)I + (E3 −E1)S+ [2E2 − (E1 +E3)]S5}

(8)where

S=

1 0 0

0 0 0

0 0 −1

0BB@1CCA (9)

and

S5 =

0 0 0

0 1 0

0 0 0

0BB@1CCA (10)

Then, sinceS5 = I − S2, Eq. (8) transforms into:

HM =E2I + 12"(q0 +q1)S− 1

2"(q0 −q −1)S2 (11)

The matrix I can be dropped from Eq. (11) by a con-venient shift of the energy origin. Taking this origin atthe energy of the intermediate levelE2, we have

HM = 12"(q0 +q1)S−

12

"(q0 −q −1)S2 (12)

Thus, much as it is usually done in the two-bodyproblem of mechanics, we have reduced the three-level problem to the separate evolution of the centerof energies and the relative energy. Note that for theharmonic oscillator case only the term depending on(q0 + q1) survives. Whenq0 Þ q1 there would then bean additional radiative mechanism to be superimposedto the usual one. Hence, two independent contributingfrequencies should be expected.

The fully quantized Hamiltonian for the three-levelmolecular system depicted in Fig. 1, for the case that

we irradiate with two beams of frequenciesqL1 andqL2, becomes

H3 ="qL1a†1a1 +"qL2a†

2a2 + 12" (q0 +q1)S�

− (q0 −q1)S2+2b a†M1 a†(N −M)

2 S8S4 −aM1 a(N −m)

2 S2S6

� ��

�13�whereb is now the dipole moment matrix element forthe total absorption of theN photons,M with energy"qL1 andN − M with energy"qL2. The operatorsSi

are the generators of the unitary groupU(3) and obey

[Si ,Sj ] = Ckij Sk (14)

in which i, j, k = 1, 2,…,9, andCkij are the structure

constants.Introducing the constants of motion ([N1,H3] =

[n2,H3] = 0)

N1 =a†1a1 +

M2

S

N2 =a†2a2 +

n−M2

S (15)

from the Heisenberg equations, we obtain finally

S=e

"(H3 −"qL1N1 −"qL2N2)Q2

RcS+Q2RrS

2 (16)

where

e =2[q0 +q1 −MqL1 − (N −M)qL2]

Q2Rc = −

e

2(q0 +q1 −MqL1) +

e

2(N −M)qL2

− lbl2A3(n1,n2) �17�

Q2Rr = −

e

2(q1 −q0) −2lbl2B3(n1,n2) (18)

with

A3(n1,n2) =n1!n2!

(n1 −M)!(n2 −N +M)!

+(n1 +M)!(n2 +N −M)!

n1!n2!

B3(n1,n2) =Mn2!

(n2 −N +M)!∑

M −1

a =0

(n1 +a)!(n1 −M +a +1)!

+(N −M)n1!

(n1 −M)!∑

N −M −1

g =0

(n2 +g)!(n2 −N +M +g +1)!

andni =a†i ai .Fig. 1. Relevant transitions in a three-level system.

61P.F. Gonza´lez-Dıaz/Journal of Molecular Structure (Theochem) 433 (1998) 59–62

The roots of the coefficients for the matricesSandS2 can be consistently interpreted, respectively, as theRabi frequency for the center of energiesQRc, and theRabi frequency for the relative energyQRr. We seethat Eqs. (15) and (16) are again radiation-intensitydependent.

ForN = M = 1 in the limitq1 =qL1 → 0,S→ jz andb → f, so that the whole three-level system reduces toan effective two-level system, andQRr becomes aconstant to be absorbed, one part,− 2lfl2, into theconventional Rabi frequency, and the other part,q0(q0 − qL1), into the counterpart of the first term inthe righthand side of Eq. (14) which contains theconstants of motion. This seems to suggest thatQRr

ultimately originates from a non-linear contribution toradiation–matter interaction through which bothradiation beams and the matter system would allcooperate. In fact, the only contributions that can sur-vive the rotating wave approximation with respect toq0 − q1 in the semiclassical treatment (where we con-sider a quantized matter system being perturbed by aclassical radiation field [13]) are those arising fromnon-linear terms which simultaneously contain bothradiation field perturbations. Thus, one should regardthe presence of the separately operating frequenciesQRc andQRr as a consequence from the full quantizedpicture. In the semiclassical treatment, possible some-what analogous effects arising from the novel non-linear Rabi frequency could still be found, thoughthey would not be expected to be visualizable asbeing driven from two independent Rabi frequencies,but rather from a more complicated single Rabiparameter.

The incidence that one could expect from the abovepredictions on multiphoton absorption of polyatomicmolecules is twofold. On one hand, the new Rabifrequency would provide us with a novel mechanismto help overcoming the molecular anharmonic barrieroperating in the few low lying levels; on the otherhand, such a Rabi frequency would play the role ofstill another control parameter in models where

dichromatic multiphoton excitation is assumed to bedriven by intermolecular generation of deterministicchaos, allowing for a more effective coupling of theactive excited levels to a harmonic reservoir, andhence leading to an increase of the multiphton absorp-tion cross section.

In summary, starting with the consideration of mul-tiphoton absorption by a two-level system, the presentreport contains the derivation of two fundamental fre-quencies in order to characterize a two-frequencylaser induced multiabsorption by a three-level system.Such frequencies are radiation-intensity dependentand correspond to the center of energy and relativeenergy, respectively.

Acknowledgements

For helpful comments and help in the calculations,the author thanks C. Sigu¨enza and M. Santos. Thisresearch was supported by DGICYT under researchprojects No. PB94-0107 and No. PB93-0139.

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