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Enhanced two-photon transitions in molecules with permanent dipole moments
Volume 144, number 2
CHEMICAL PHYSICS LETTERS
19 February 1988
ENHANCED TWO-PHOTON TRANSITIONS IN MOLECULES WITH PERMANENT DIPOLE MOMENTS Benjamin E. SCHARF Department of Chemistry, Ben-Gurion Universityof The Negev, Beer-Sheva 84105, Israel
and Yehuda B. BAND Department of Chemistry Ben-Gurion Universityof The Negev, Beer-Sheva 84105, Israel and Allied-Signal, Inc., Corporate Technology Center, Morristown, NJ 07960, USA Received 8 October 1987
For radiative processes involving two photons of different color in the limit when one of the photon frequencies is small, we show that the rate becomes very large due to a unique resonance. The two-photon rate becomes comparable with one-photon transition rates, even for low radiation field intensities. Applications of this phenomenon are discussed.
1. Introduction Two-photon radiative processes (simultaneous two-photon absorption, emission, and Raman processes) are described in terms of second-order perturbation theory involving the photon-matter interaction Hamiltonian, X. The transition rate expression for such processes from an initial state i to a final state f is given in terms of an infinite sum over all intermediate states {e} of products of matrix elements (fIXI e) (elX1 i) times a suitable energy denominator. This sum contains contributions from all two-step processes involving three-level systems; a virtual transition from state i to state e is followed by a virtual transition from state e to f. Under certain conditions the description of two-photon transitions will also contain contributions from two-step processes involving only two-level systems when the E-r operator form of the photon-matter interaction is used, i.e. terms of the form ( f 1r I i) (i I r (i) and ( f I r If) ( f Ir I i) contribute [ 11. Clearly, a non-vanishing permanent dipole moment (g Irl g) is necessary of such terms to contribute. In this Letter we show that when (i) the initial and final state have permanent dipole moments and their difference is substantial, and (ii) the transition dipole moment between the initial and final states is appreciable, the above mentioned terms become the dominant terms in the transition rate expression for radiative processes involving two photons of different color in the limit when one of the photon frequencies is small, and the rate becomes very large! This occurs because the energy denominators appearing in the two dominating terms of the second-order perturbation expression become small, i.e. a resonance condition is met. In fact, the two-photon rate becomes comparable with one-photon transition rates, even for low radiation field intensities. Thus, two-photon absorption experiments become feasible even with low incident power levels and/or low sample concentrations. Moreover, the gain for two-photon-induced emission may become sufficiently large, allowing lasing to ensue. This process involves amplification of two photon beams of different color injected into a molecular sample containing an appropriate excited state population inversion. The works quoted in ref. [ I] did not consider the limit in which one of the photon frequencies is small (but dealt with two photons of equal frequency) and therefore did not recognize the strong enhancement of the rates of two-photon processes in 0 009-2614/88/$ 03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
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molecules with permanent dipole moments due to a unique resonance occurring for these cases. The outline of the Letter is as follows. Section 2 reviews the various forms of second-order perturbation theory for radiative transitions. Section 3 contains the analysis of the special case when the initial or final states possess permanent dipole moments’and one of the two photon frequencies is small. Section 4 describesa dressed state picture for treating these resonance cases. In section 5 we outline several applications of the resonance cases described here.
2. Momentum and position forms of second-orderradiative transitions The second-order perturbation theory expression for the rate of transition from level i to level f can be written in the form [ 21 (atomic units are used throughout) Wfi=2xI(flMIi)12Pr,
(1)
where the matrix element ( f (MI i) involves a sum over all possible intermediate states e (Wli)
=C
e
(
(elX2li)
Ei-E,+it
+WGli>
I
(2)
Ei -E,+ie
and pf is the density of final states. Here X1 and X2 are the interaction operators responsible for the transition (note that X, can be identical to X2 when only one interaction operator is responsible for the transition). When the transition is due to electric dipole radiative processes the interaction operators & are given by X,= -Ak -plm where we take the components of Ak to be AL’), A~+)=c[(N~+~)/~w~V]“~~~,
A~-)=c(N~/~co~T/)~‘~~~,
(3)
for emission and absorption respectively. Here c is the speed of light, V is the volume, and Nk and cukare the number of photons and frequency of mode k present in the radiation field. For Raman scattering (involving absorption of radiation of mode 2 and emission of radiation of mode 1 ), or for two-photon absorption, the second-order perturbation theory matrix elements take the following forms respectively:
(fWl0
‘(-&)‘C(
(flAI+).pJe)(elA~-)~pli) E, +02 -E,+ic
+ (flAb-)~ple)(elA~‘)g~li) Ei-W, -E, tit
(flA\-‘.ple) (elA$-‘*pIi) Ei+oz-E,tie
+ (flAl-)*ile> Ei+Ol
’
(4)
We shall not pause to present the derivation of the two-photon-induced emission case since the changes that need to be made are obvious. The forms given in eq. (4) contain matrix elements of the momentum operator. Clearly, only states Ie) with non-vanishing momentum matrix elements with states (i> and )f) contribute to the sum in eq. (4). It is sometimes useful to convert the momentum forms of electric dipole matrix elements to the position forms. in the present application this is shown to be particularly true. Using the relations (j lp )k) = - im( E,-E,J (j I r I k) , and -EL’)= +w$o4~*~, the following relation between matrix elements can be easily derived: (jl -Akop/mc) n) = ~wj,/w, (jl E,*r/mcl n). Here o,, is the energy difference between statesj and II, and u)k is the energy of the photon in radiation mode k. When substituted into eq. (4) we obtain
lrle)(elEl-‘*rli} E,tw,-E,tie
+ (flee-).rle)(elEI+).rli} Ei-01 -E,tie
’
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Peticolas [ 31 showed us how to transform the expression in eq. (5) into simpler expressions by adding an additional sum which equals zero to the right-hand sides of the equations to obtain the results (for Raman, two-photon absorption, and two-photon emission respectively)
(flEI+).rle)(elEI+)*rli) Ei-W,-Ee+ie
+ (flee-)*rle)(elEI+).rli) &-CO* -E,+iC
*
(6)
The expressions in eq. (6) are identical to those we would obtain if instead of using the interactions X,= -Ak~plm we had used Xk= - Ek.r in the second-order perturbation theory expression, eq. (2). When the energy denominator of the terms in eq. (8) becomes very small, E should be replaced by the inverse of the T2 decay rate, rei [ 41. The three forms of second-order matrix elements are given in eqs. (4)) (5) and (6). If a complete set of states 1e) and exact wavefunctions f and i are used, these three expressions must yield identical results. However, if approximations are made, differences between the results of these three expressions may result. For example, it is easier to accurately numerically evaluate matrix elements involving the position operator than matrix elements involving the momentum operator (which requires that a numerical derivative be calculated). What about differences between results using eqs. (5) and (6) which both involve the position operator? If a truncation of the complete set of states {e} is performed, the results using eqs. (5) and (6) are not necessarily identical. Eq. (6) is more often used in the literature. The reason for this is twofold: (1) the sum over all states Ie} in eq. (6) yields the Greens’ function for the intermediate state Hamiltonian [ 5-71 Gi(q,q’)E(qI(E$ie-H,)-‘1q’),
(7)
G~(~,~‘)=(~l[CIe~(~-~,~i~)-1(eIlIq’)=2~~~(~,~’)~~(q~)h~(q,),
(8)
where SE(q) and hi (q) are solutions to the Hamiltonian H, with eigenvalue E, and q, (q<) is the larger (smaller) of q, q’; (2) even if the spectral decomposition (sum over all e) is retained, the sum in eq. (6) seems to converge faster and there is no need to incorporate the terms o~,w,/o, w2 in the sum over e thus simplifying the calculation. The form given in eq. (6) was originally used by Goppert-Meyer for two-photon absorption in 193 1 [ 8 1. We shall see that for the applications discussed in section 3, there is an additional excellent reason for using eq. (6). In addition to the second-order perturbation terms contributing to the f+i transition, a first-order perturbation term arising from the interaction A(r, t)*/2mc* also contributes. Expanding this interaction in a multipole expansion, the lowest-order multipole term contributing to the first-order matrix element, A#), gives M~1)=(2mc2)-‘A,*Az
i(k, +k2)*(flrli)
.
(9)
For the processes considered here, this term is much smaller than terms originating from second-order perturbation theory when w, or CL)* are in the visible or IR region of the spectrum since 1(k, +k2 ) * ( f] r 1i) 1s 1.
3. Second-order transitions in molecules with permanent dipole moments Let us consider two-photon processes where two different colors are used in the limit when one of the photon frequencies, say CO,, becomes small. In this case, two terms in the expressions given in eq. (6) become large 167
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and dominate the expression for the transition rate when the initial or final states possess permanent dipole moments. This simplification only occurs if the forms in eq. (6) are used. The momentum form, eq. (4)) or the position form in eq. (5), do not yield two dominant terms and therefore it is clearly preferential to use eq. (6) in these cases. The rate of change of population of the final level is given for two-photon absorption, stimulated two-photon emission, and stimulated Raman scattering by dNfldt=NiWf,=4n3C-‘Ni(~,wI)
(&o~)~S~I~)
(10)
where the matrix element Sfi is simply related to & of eq. (6) by substituting el, e2 for El, E2, &i=Mi~lE, I IE2 I ,
(11)
and #,, & are the photon fluxes of fields 1,2 in units of photons/(areaxtime) respectively. For spontaneous twd-photon emission, IS, )’ also appears in the expression for the rate of change of Nf but the other multiplicative factors on the right-hand side of eq. (10) need modification. In what follows we present the expressions for & for two-photon absorption, two-photon emission, and Raman transitions. We consider those terms in the expression for two-photon absorption obtained in eq. (6) when the intermediate state e is taken to be i and f. These terms yield s, = (fle,
~40
(fle2*rli)
Ei+w,-E,+k
= (fle,-vlf)(fle~.rli).+ -wl +it
+ (fle2*4i>
(ilet
Ai)
Ei+W, -Ei+ie
(f(ez*rli)(ile,*rli> w, +ie
(12) ,
where we have used conservation of energy, Ei +w2 +w, -E,=O, to obtain the second equality in eq. (12). Using eq. (6) for two-photon emission and the conservation of energy relation Ei -co2 -co, - E,=O, exactly the same equation for S, as that deduced in eq. (12) is obtained, whereas for Raman transitions the conservation of energy relation Ei + w2 -cc) 1- Ef = 0 yields ( - 1) times the right-hand side of eq. (12).
4. Resonance limit - dressed state treatment
In the discussion above we used second-order perturbation theory to calculate the transition rates. However, we observed that these rates become large in the limit that o1 is small and the initial or final states have a permanent dipole moment. Clearly, the regime of validity of the perturbation expression is ultimately exceeded when the rate becomes sufficiently large. If inclusion of the decay rate rd is not enough to hold back the rate from exceeding the regime of validity of perturbation theory [ 41, we must consider the transition to infinite order in the field strengths. To correctly treat this problem to all orders in both radiation fields 1 and 2 one can dress states i and f with both fields using the dressed state formalism developed by Jaynes and Cummings and others [ 91. Here, we outline this dressed state procedure as applied to the processes considered. A full treatment of the dressed state procedure is outside the scope of this Letter. For Raman scattering, we couple state (i, n2, n, ) with states Ie, n2 - 1, n, ) and I e, n2, n, + 1) through the interaction of the matter-radiation interaction Hamiltonian H2 and H, respectively, and these states are in turn coupled with the final state If, n2 - 1, n, + 1) via H, and Hz respectively. Defining the interaction matrix elements tsl,=(i
lG*=
n2, ItI 1x21~ n2-L
n1 >,
(i, n2, nl I4 le, n2, n,+ I>,
$2;=(e,n2,nl+11X21f,n2-l,n,tl), QSf=(i,n,-1,n,IX,Ie,n2-l,n,+l),
the Hamiltonian can be written in terms of the dressed state basis functions as
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Ii,
n2,
h>
Ei-E I% &?;* 0
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CHEMICAL PHYSICS LETTERS
le,
n2-1,
tn, E,-E-w, 0 fG
h>
le,
n2,
4+1)
$2;+ 0 E,-E+w, tf&
If,
n2-1,
n1+1)
0 fsl: Q& Ef-E-co2
+w,
I-
(14)
One can dress the molecular levels with the field at frequency o, to obtain dressed states 1In, ) and 1 > and dressed states ( Fn2 + 1> and 1En2 + 1) which are linear combinations of 1fn, + 1n l - 1) and Ien2 -I1n, ). One can then calculate the dressed state transition matrix element for the Raman process I Fn, + 11X2 I In, > in terms of the mixing angles 29,and 0; defined by tan(28,)=52,l(E,-E,-w,) and tan(219;)=Q;l(E,-Ef-w,) respectively and the Rabi frequencies Sz, and Sz;. If this dressing of the states with the field at frequency w, is not sufficient to converge the transition matrix element, a dressing of the states with respect to the field at frequency wz can be performed, however, the terms left out of the rotating wave approximation may have to be reincluded since wz is small. Details of this procedure will be presented elsewhere [ lo].
1En2) which are linear combinations of I inzn, > and I tw2fl, -
5. Applications Enhancement in the rate of two-photon transitions is expected when: (i) the initial and final state have permanent dipole moments whose difference is substantial, and (ii) the transition dipole moment between the initial and final states is appreciable. These enhancement effects will increase dramatically when the frequency of one of the photons, say 02, is very close to the frequency of the two-photon transition, and w , is small. For electronic transitions when the small frequency w 1is of the order of 1 cm- ’ and the difference in the permanent dipole moments of the initial and final states is z 1OV cm (4.8 x lo-’ esu), an enhancement in the two-photon rate by a factor of lo9 may occur as compared with the two-photon rate in systems where conditions (i) and (ii) do not hold. If the difference in the permanent dipole moments is of order 3 X lo-’ cm (for instance paranitroaniline and similar compounds) the enhancement factor is 10 ‘O. In molecular systems these resonance enhancement effects are this large because of the unity vibrational overlap factors in the (i I r Ii ) and ( f Ir I f) dipole matrix elements. Notice also that it is not the derivative of the dipole moment with respect to internuclear coordinates but rather the dipole moment itself which enters. The enhancement varies with w, as WI-2 when the intensities of the two beams (or their product) is kept constant. These estimates, and those below as well, presume that saturation of the two-photon transitions does not take place so that breakdown of the second-order perturbation nature of the transition does not occur and the dressed state picture is not necessary. We now compare the rate of the two-photon processes with one-photon processes. For electronic transitions when resonance conditions do not prevail and the intensity of one of the beams equals that in the one-photon experiment while the intensity of the second beam is 1 GW/cm’, the rate of the two-photon process will be x 10m4times slower than the rate of the one-photon process [ 31. Considering a resonance enhancement factor of lOlo, the rate of the two-photon process may match that of the one-photon process when the intensity of the w 1= 1 cm-’ beam is equal or larger than 1O3W/cm*. If w , = 100 cm-‘, an intensity of > lo7 W/cm2 is needed to match the rate of the one-photon process. One may use the resonance-induced two-photon processes to carry out a two-photon absorption experiment, using laser-induced emission or multiphoton ionization techniques, which isnot masked by competing onephoton processes. Such an experiment may enable measurement of the variation of the anomalous permanent electric dipole moment of the final vibronic states in A-E or A+T electronic transitions [ 1I- 131. There are 169
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molecular systems for which an anomalous permanent electric dipole moment may arise in their degenerate electronic states, although no permanent electric dipole moments are allowed in their normal non-degenerate electronic states due to symmetry prohibitions. Examples of such systems are the E states in molecules of D3,,, Dzd, DAdand Ded symmetry or T states in molecules of Td symmetry. These ano.malous electric dipole moments which may be as large as usual permanent electric dipole moments do not result via distortion in nuclear configuration. However, Jahn-Teller, Renner-Teller and mixed cross quadratic nuclear potential terms [ 14,15 ] may lead to strong variations of the anomalous dipole moments in the various vibronic components [ 1l-l 51. The strength of the two-photon vibronic transitions will be determined by the vibronic factors which are relevant for the direct one-photon transition and by the vibronic anomalous dipole moment of the final vibronic level in the degenerate electronic state. This is because in these systems no electronic dipole moment occurs in the non-degenerate electronic state. There are systems where an anomalous and a regular permanent electric dipole moment can coexist. Strong resonance enhancement effects may occur for two-photon vibrational transitions within electronically degenerate states possessing anomalous dipole moments. The rates of one-photon and two-photon absorption processes in these systems may be comparable to the rates of these processes in electronic transitions when the same intensities and resonance conditions prevail. This is a remarkable result since the rate of one-photon vibrational transitions for the Jahn-Teller active modes in degenerate electronic states may be lo3 times faster than in non-degenerate electronic states for cases where anomalous electric dipole moments prevail in the degenerate electronic state. This results because the vibrational transitions in normal non-degenerate electronic cases derive their intensity from the derivative of the permanent electric dipole moment with respect to the relevant vibrational coordinate, whereas in the degenerate case the intensity may derive from the (larger) anomalous permanent electric dipole moment [ 1l- 131. Furthermore, the intensity of simultaneous two-photon vibrational transitions for harmonic potentials in non-degenerate electronic states vanishes [ 11. These restrictions do not hold for two-photon vibrational transitions in degenerate electronic states which possess anomalous dipole moments and are Jahn-Teller active [ 131. In such cases two-photon transitions involving even-quantum and odd-quantum jumps may ensue and undergo strong enhancement effects. The O-l and O-2 resonance-enhanced two-photon transitions may occur. Resonance-enhanced two-photon-induced emission may be used to induce two-photon lasing. Amplified induced emissions of two laser beams with frequencies w , and w2 may occur under proper population inversion between the initial excited and final states associated with a resonance-enhanced transition whose frequency is w , t oz. Similarly, an amplification in CC)~ may be obtained in a resonance-enhanced induced Raman process where the transition frequency equals IX,--o 2. In these processes the intensities of the laser beams must be sufficiently large so that these processes compete with the relevant one-photon spontaneous and induced emissions. These two-photon lasing processes are feasible since under proper experimental conditions the radiative rate of the two-photon-induced emission under resonance conditions and proper intensities may match and exceed the rate of the competing one-photon-induced emission processes. In dye lasers one often uses injection seeding by a low intensity monochromatic beam to obtain single longitudinal mode output. We suggest that a similar technique might be developed using injection of two monochromatic beams to produce amplification of the two beams. The applicability of these ideas should be examined in different regions of the spectrum ranging from UV, visible through the IR, microwave, and down to the rf.
Acknowledgement This work was supported in part by a grant from the US-Israel Binational Science Foundation.
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References [ I] GE Thomas and W.J. M&h, Mol. Phys. 46 (1982) 951; W.J. Meath and EA. Power, J. Phys. B17 (1984) 763; Mol. Phys. 5 1 ( 1984) 585; B. Dick and G. Hohlneicher, J. Chem. Phys. 76 (1982) 5755. [ 21 W.M. McClain and R.A. Harris, in: Excited states, Vol.3, ed. E.C. Lim (Academic Pres, New York, 1977) pp. 2-56; B. Honig, J. Jortner and A. Szoke, J. Chem. Phys. 46 (1967) 2714. [3] W.L. Peticolas, J. Chem. Phys. 42 (1965) 4164; Ann. Rev. Phys. Chem. 18 (1967) 233. [ 41 N. Bloembergen, Nonlinear optics (Academic Press, New York, 1965) ch. 2. [5] Y.B. Band and B.M. Aizenbud, Chem. Phys. Letters 77 (1981) 49. [ 6) Y.B. Band and B.M. Aizenbud, Chem. Phys. Letters 79 ( 1981) 244. [ 71 S.J. Singer, S. Lee, K.F. Freed and Y.B. Band, J. Chem. Phys., to be published. [ 8 ] M. Goppert-Meyer, Ann. Physik (Leipzig) [ 51 9 ( 1931) 273. [ 91 E.T. Jaynes and F.W. Cummings, Proc. IEEE 5 1 (1963) 89; T.L. Knights and P.W. Milloni, Phys. Rept. 66 (1980) 22; C. Cohen-Tannoudji and S. Reynaud, J. Phys. Al0 ( 1977) 345; F.H. Mies and Y. Ben Aryeh, J. Chem. Phys. 74 (1981) 53. [ IO] F.H. Mies and Y.B. Band, to be published. [ 111 M.S. Child and H.C. Longuet-Higgins, Phil. Trans. Roy. Sot. (London) A254 (1961) 259. [ 121 B. Scharfand Y.B. Band, J. Chem. Phys. 79 (1983) 3175,3182. [ 131 B. ScharfandT.A. Miller, J. Chem. Phys. 84 (1986) 561. [ 141 B. Schatf, Chem. Phys. Letters 96 (1983) 89; 98 (1983) 81; 99 (1983) 350; 102 (1983) 184; 114 (1985) 291. [ 151 R.A. Kennedy, T.A. Miller and B. Scharf, J. Chem. Phys. 85 (1986) 1336.
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ENHANCED TWO-PHOTON TRANSITIONS IN MOLECULES WITH PERMANENT DIPOLE MOMENTS Benjamin E. SCHARF Department of Chemistry, Ben-Gurion Universityof The Negev, Beer-Sheva 84105, Israel
and Yehuda B. BAND Department of Chemistry Ben-Gurion Universityof The Negev, Beer-Sheva 84105, Israel and Allied-Signal, Inc., Corporate Technology Center, Morristown, NJ 07960, USA Received 8 October 1987
For radiative processes involving two photons of different color in the limit when one of the photon frequencies is small, we show that the rate becomes very large due to a unique resonance. The two-photon rate becomes comparable with one-photon transition rates, even for low radiation field intensities. Applications of this phenomenon are discussed.
1. Introduction Two-photon radiative processes (simultaneous two-photon absorption, emission, and Raman processes) are described in terms of second-order perturbation theory involving the photon-matter interaction Hamiltonian, X. The transition rate expression for such processes from an initial state i to a final state f is given in terms of an infinite sum over all intermediate states {e} of products of matrix elements (fIXI e) (elX1 i) times a suitable energy denominator. This sum contains contributions from all two-step processes involving three-level systems; a virtual transition from state i to state e is followed by a virtual transition from state e to f. Under certain conditions the description of two-photon transitions will also contain contributions from two-step processes involving only two-level systems when the E-r operator form of the photon-matter interaction is used, i.e. terms of the form ( f 1r I i) (i I r (i) and ( f I r If) ( f Ir I i) contribute [ 11. Clearly, a non-vanishing permanent dipole moment (g Irl g) is necessary of such terms to contribute. In this Letter we show that when (i) the initial and final state have permanent dipole moments and their difference is substantial, and (ii) the transition dipole moment between the initial and final states is appreciable, the above mentioned terms become the dominant terms in the transition rate expression for radiative processes involving two photons of different color in the limit when one of the photon frequencies is small, and the rate becomes very large! This occurs because the energy denominators appearing in the two dominating terms of the second-order perturbation expression become small, i.e. a resonance condition is met. In fact, the two-photon rate becomes comparable with one-photon transition rates, even for low radiation field intensities. Thus, two-photon absorption experiments become feasible even with low incident power levels and/or low sample concentrations. Moreover, the gain for two-photon-induced emission may become sufficiently large, allowing lasing to ensue. This process involves amplification of two photon beams of different color injected into a molecular sample containing an appropriate excited state population inversion. The works quoted in ref. [ I] did not consider the limit in which one of the photon frequencies is small (but dealt with two photons of equal frequency) and therefore did not recognize the strong enhancement of the rates of two-photon processes in 0 009-2614/88/$ 03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
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molecules with permanent dipole moments due to a unique resonance occurring for these cases. The outline of the Letter is as follows. Section 2 reviews the various forms of second-order perturbation theory for radiative transitions. Section 3 contains the analysis of the special case when the initial or final states possess permanent dipole moments’and one of the two photon frequencies is small. Section 4 describesa dressed state picture for treating these resonance cases. In section 5 we outline several applications of the resonance cases described here.
2. Momentum and position forms of second-orderradiative transitions The second-order perturbation theory expression for the rate of transition from level i to level f can be written in the form [ 21 (atomic units are used throughout) Wfi=2xI(flMIi)12Pr,
(1)
where the matrix element ( f (MI i) involves a sum over all possible intermediate states e (Wli)
=C
e
(
Ei-E,+it
+
I
(2)
Ei -E,+ie
and pf is the density of final states. Here X1 and X2 are the interaction operators responsible for the transition (note that X, can be identical to X2 when only one interaction operator is responsible for the transition). When the transition is due to electric dipole radiative processes the interaction operators & are given by X,= -Ak -plm where we take the components of Ak to be AL’), A~+)=c[(N~+~)/~w~V]“~~~,
A~-)=c(N~/~co~T/)~‘~~~,
(3)
for emission and absorption respectively. Here c is the speed of light, V is the volume, and Nk and cukare the number of photons and frequency of mode k present in the radiation field. For Raman scattering (involving absorption of radiation of mode 2 and emission of radiation of mode 1 ), or for two-photon absorption, the second-order perturbation theory matrix elements take the following forms respectively:
(fWl0
‘(-&)‘C(
(flAI+).pJe)(elA~-)~pli) E, +02 -E,+ic
+ (flAb-)~ple)(elA~‘)g~li) Ei-W, -E, tit
(flA\-‘.ple) (elA$-‘*pIi) Ei+oz-E,tie
+ (flAl-)*ile>
’
(4)
We shall not pause to present the derivation of the two-photon-induced emission case since the changes that need to be made are obvious. The forms given in eq. (4) contain matrix elements of the momentum operator. Clearly, only states Ie) with non-vanishing momentum matrix elements with states (i> and )f) contribute to the sum in eq. (4). It is sometimes useful to convert the momentum forms of electric dipole matrix elements to the position forms. in the present application this is shown to be particularly true. Using the relations (j lp )k) = - im( E,-E,J (j I r I k) , and -EL’)= +w$o4~*~, the following relation between matrix elements can be easily derived: (jl -Akop/mc) n) = ~wj,/w, (jl E,*r/mcl n). Here o,, is the energy difference between statesj and II, and u)k is the energy of the photon in radiation mode k. When substituted into eq. (4) we obtain
lrle)(elEl-‘*rli} E,tw,-E,tie
+ (flee-).rle)(elEI+).rli} Ei-01 -E,tie
’
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Peticolas [ 31 showed us how to transform the expression in eq. (5) into simpler expressions by adding an additional sum which equals zero to the right-hand sides of the equations to obtain the results (for Raman, two-photon absorption, and two-photon emission respectively)
(flEI+).rle)(elEI+)*rli) Ei-W,-Ee+ie
+ (flee-)*rle)(elEI+).rli) &-CO* -E,+iC
*
(6)
The expressions in eq. (6) are identical to those we would obtain if instead of using the interactions X,= -Ak~plm we had used Xk= - Ek.r in the second-order perturbation theory expression, eq. (2). When the energy denominator of the terms in eq. (8) becomes very small, E should be replaced by the inverse of the T2 decay rate, rei [ 41. The three forms of second-order matrix elements are given in eqs. (4)) (5) and (6). If a complete set of states 1e) and exact wavefunctions f and i are used, these three expressions must yield identical results. However, if approximations are made, differences between the results of these three expressions may result. For example, it is easier to accurately numerically evaluate matrix elements involving the position operator than matrix elements involving the momentum operator (which requires that a numerical derivative be calculated). What about differences between results using eqs. (5) and (6) which both involve the position operator? If a truncation of the complete set of states {e} is performed, the results using eqs. (5) and (6) are not necessarily identical. Eq. (6) is more often used in the literature. The reason for this is twofold: (1) the sum over all states Ie} in eq. (6) yields the Greens’ function for the intermediate state Hamiltonian [ 5-71 Gi(q,q’)E(qI(E$ie-H,)-‘1q’),
(7)
G~(~,~‘)=(~l[CIe~(~-~,~i~)-1(eIlIq’)=2~~~(~,~’)~~(q~)h~(q,),
(8)
where SE(q) and hi (q) are solutions to the Hamiltonian H, with eigenvalue E, and q, (q<) is the larger (smaller) of q, q’; (2) even if the spectral decomposition (sum over all e) is retained, the sum in eq. (6) seems to converge faster and there is no need to incorporate the terms o~,w,/o, w2 in the sum over e thus simplifying the calculation. The form given in eq. (6) was originally used by Goppert-Meyer for two-photon absorption in 193 1 [ 8 1. We shall see that for the applications discussed in section 3, there is an additional excellent reason for using eq. (6). In addition to the second-order perturbation terms contributing to the f+i transition, a first-order perturbation term arising from the interaction A(r, t)*/2mc* also contributes. Expanding this interaction in a multipole expansion, the lowest-order multipole term contributing to the first-order matrix element, A#), gives M~1)=(2mc2)-‘A,*Az
i(k, +k2)*(flrli)
.
(9)
For the processes considered here, this term is much smaller than terms originating from second-order perturbation theory when w, or CL)* are in the visible or IR region of the spectrum since 1(k, +k2 ) * ( f] r 1i) 1s 1.
3. Second-order transitions in molecules with permanent dipole moments Let us consider two-photon processes where two different colors are used in the limit when one of the photon frequencies, say CO,, becomes small. In this case, two terms in the expressions given in eq. (6) become large 167
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and dominate the expression for the transition rate when the initial or final states possess permanent dipole moments. This simplification only occurs if the forms in eq. (6) are used. The momentum form, eq. (4)) or the position form in eq. (5), do not yield two dominant terms and therefore it is clearly preferential to use eq. (6) in these cases. The rate of change of population of the final level is given for two-photon absorption, stimulated two-photon emission, and stimulated Raman scattering by dNfldt=NiWf,=4n3C-‘Ni(~,wI)
(&o~)~S~I~)
(10)
where the matrix element Sfi is simply related to & of eq. (6) by substituting el, e2 for El, E2, &i=Mi~lE, I IE2 I ,
(11)
and #,, & are the photon fluxes of fields 1,2 in units of photons/(areaxtime) respectively. For spontaneous twd-photon emission, IS, )’ also appears in the expression for the rate of change of Nf but the other multiplicative factors on the right-hand side of eq. (10) need modification. In what follows we present the expressions for & for two-photon absorption, two-photon emission, and Raman transitions. We consider those terms in the expression for two-photon absorption obtained in eq. (6) when the intermediate state e is taken to be i and f. These terms yield s, = (fle,
~40
(fle2*rli)
Ei+w,-E,+k
= (fle,-vlf)(fle~.rli).+ -wl +it
+ (fle2*4i>
(ilet
Ai)
Ei+W, -Ei+ie
(f(ez*rli)(ile,*rli> w, +ie
(12) ,
where we have used conservation of energy, Ei +w2 +w, -E,=O, to obtain the second equality in eq. (12). Using eq. (6) for two-photon emission and the conservation of energy relation Ei -co2 -co, - E,=O, exactly the same equation for S, as that deduced in eq. (12) is obtained, whereas for Raman transitions the conservation of energy relation Ei + w2 -cc) 1- Ef = 0 yields ( - 1) times the right-hand side of eq. (12).
4. Resonance limit - dressed state treatment
In the discussion above we used second-order perturbation theory to calculate the transition rates. However, we observed that these rates become large in the limit that o1 is small and the initial or final states have a permanent dipole moment. Clearly, the regime of validity of the perturbation expression is ultimately exceeded when the rate becomes sufficiently large. If inclusion of the decay rate rd is not enough to hold back the rate from exceeding the regime of validity of perturbation theory [ 41, we must consider the transition to infinite order in the field strengths. To correctly treat this problem to all orders in both radiation fields 1 and 2 one can dress states i and f with both fields using the dressed state formalism developed by Jaynes and Cummings and others [ 91. Here, we outline this dressed state procedure as applied to the processes considered. A full treatment of the dressed state procedure is outside the scope of this Letter. For Raman scattering, we couple state (i, n2, n, ) with states Ie, n2 - 1, n, ) and I e, n2, n, + 1) through the interaction of the matter-radiation interaction Hamiltonian H2 and H, respectively, and these states are in turn coupled with the final state If, n2 - 1, n, + 1) via H, and Hz respectively. Defining the interaction matrix elements tsl,=(i
lG*=
n2, ItI 1x21~ n2-L
n1 >,
(i, n2, nl I4 le, n2, n,+ I>,
$2;=(e,n2,nl+11X21f,n2-l,n,tl), QSf=(i,n,-1,n,IX,Ie,n2-l,n,+l),
the Hamiltonian can be written in terms of the dressed state basis functions as
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Ii,
n2,
h>
Ei-E I% &?;* 0
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CHEMICAL PHYSICS LETTERS
le,
n2-1,
tn, E,-E-w, 0 fG
h>
le,
n2,
4+1)
$2;+ 0 E,-E+w, tf&
If,
n2-1,
n1+1)
0 fsl: Q& Ef-E-co2
+w,
I-
(14)
One can dress the molecular levels with the field at frequency o, to obtain dressed states 1In, ) and 1 > and dressed states ( Fn2 + 1> and 1En2 + 1) which are linear combinations of 1fn, + 1n l - 1) and Ien2 -I1n, ). One can then calculate the dressed state transition matrix element for the Raman process I Fn, + 11X2 I In, > in terms of the mixing angles 29,and 0; defined by tan(28,)=52,l(E,-E,-w,) and tan(219;)=Q;l(E,-Ef-w,) respectively and the Rabi frequencies Sz, and Sz;. If this dressing of the states with the field at frequency w, is not sufficient to converge the transition matrix element, a dressing of the states with respect to the field at frequency wz can be performed, however, the terms left out of the rotating wave approximation may have to be reincluded since wz is small. Details of this procedure will be presented elsewhere [ lo].
1En2) which are linear combinations of I inzn, > and I tw2fl, -
5. Applications Enhancement in the rate of two-photon transitions is expected when: (i) the initial and final state have permanent dipole moments whose difference is substantial, and (ii) the transition dipole moment between the initial and final states is appreciable. These enhancement effects will increase dramatically when the frequency of one of the photons, say 02, is very close to the frequency of the two-photon transition, and w , is small. For electronic transitions when the small frequency w 1is of the order of 1 cm- ’ and the difference in the permanent dipole moments of the initial and final states is z 1OV cm (4.8 x lo-’ esu), an enhancement in the two-photon rate by a factor of lo9 may occur as compared with the two-photon rate in systems where conditions (i) and (ii) do not hold. If the difference in the permanent dipole moments is of order 3 X lo-’ cm (for instance paranitroaniline and similar compounds) the enhancement factor is 10 ‘O. In molecular systems these resonance enhancement effects are this large because of the unity vibrational overlap factors in the (i I r Ii ) and ( f Ir I f) dipole matrix elements. Notice also that it is not the derivative of the dipole moment with respect to internuclear coordinates but rather the dipole moment itself which enters. The enhancement varies with w, as WI-2 when the intensities of the two beams (or their product) is kept constant. These estimates, and those below as well, presume that saturation of the two-photon transitions does not take place so that breakdown of the second-order perturbation nature of the transition does not occur and the dressed state picture is not necessary. We now compare the rate of the two-photon processes with one-photon processes. For electronic transitions when resonance conditions do not prevail and the intensity of one of the beams equals that in the one-photon experiment while the intensity of the second beam is 1 GW/cm’, the rate of the two-photon process will be x 10m4times slower than the rate of the one-photon process [ 31. Considering a resonance enhancement factor of lOlo, the rate of the two-photon process may match that of the one-photon process when the intensity of the w 1= 1 cm-’ beam is equal or larger than 1O3W/cm*. If w , = 100 cm-‘, an intensity of > lo7 W/cm2 is needed to match the rate of the one-photon process. One may use the resonance-induced two-photon processes to carry out a two-photon absorption experiment, using laser-induced emission or multiphoton ionization techniques, which isnot masked by competing onephoton processes. Such an experiment may enable measurement of the variation of the anomalous permanent electric dipole moment of the final vibronic states in A-E or A+T electronic transitions [ 1I- 131. There are 169
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molecular systems for which an anomalous permanent electric dipole moment may arise in their degenerate electronic states, although no permanent electric dipole moments are allowed in their normal non-degenerate electronic states due to symmetry prohibitions. Examples of such systems are the E states in molecules of D3,,, Dzd, DAdand Ded symmetry or T states in molecules of Td symmetry. These ano.malous electric dipole moments which may be as large as usual permanent electric dipole moments do not result via distortion in nuclear configuration. However, Jahn-Teller, Renner-Teller and mixed cross quadratic nuclear potential terms [ 14,15 ] may lead to strong variations of the anomalous dipole moments in the various vibronic components [ 1l-l 51. The strength of the two-photon vibronic transitions will be determined by the vibronic factors which are relevant for the direct one-photon transition and by the vibronic anomalous dipole moment of the final vibronic level in the degenerate electronic state. This is because in these systems no electronic dipole moment occurs in the non-degenerate electronic state. There are systems where an anomalous and a regular permanent electric dipole moment can coexist. Strong resonance enhancement effects may occur for two-photon vibrational transitions within electronically degenerate states possessing anomalous dipole moments. The rates of one-photon and two-photon absorption processes in these systems may be comparable to the rates of these processes in electronic transitions when the same intensities and resonance conditions prevail. This is a remarkable result since the rate of one-photon vibrational transitions for the Jahn-Teller active modes in degenerate electronic states may be lo3 times faster than in non-degenerate electronic states for cases where anomalous electric dipole moments prevail in the degenerate electronic state. This results because the vibrational transitions in normal non-degenerate electronic cases derive their intensity from the derivative of the permanent electric dipole moment with respect to the relevant vibrational coordinate, whereas in the degenerate case the intensity may derive from the (larger) anomalous permanent electric dipole moment [ 1l- 131. Furthermore, the intensity of simultaneous two-photon vibrational transitions for harmonic potentials in non-degenerate electronic states vanishes [ 11. These restrictions do not hold for two-photon vibrational transitions in degenerate electronic states which possess anomalous dipole moments and are Jahn-Teller active [ 131. In such cases two-photon transitions involving even-quantum and odd-quantum jumps may ensue and undergo strong enhancement effects. The O-l and O-2 resonance-enhanced two-photon transitions may occur. Resonance-enhanced two-photon-induced emission may be used to induce two-photon lasing. Amplified induced emissions of two laser beams with frequencies w , and w2 may occur under proper population inversion between the initial excited and final states associated with a resonance-enhanced transition whose frequency is w , t oz. Similarly, an amplification in CC)~ may be obtained in a resonance-enhanced induced Raman process where the transition frequency equals IX,--o 2. In these processes the intensities of the laser beams must be sufficiently large so that these processes compete with the relevant one-photon spontaneous and induced emissions. These two-photon lasing processes are feasible since under proper experimental conditions the radiative rate of the two-photon-induced emission under resonance conditions and proper intensities may match and exceed the rate of the competing one-photon-induced emission processes. In dye lasers one often uses injection seeding by a low intensity monochromatic beam to obtain single longitudinal mode output. We suggest that a similar technique might be developed using injection of two monochromatic beams to produce amplification of the two beams. The applicability of these ideas should be examined in different regions of the spectrum ranging from UV, visible through the IR, microwave, and down to the rf.
Acknowledgement This work was supported in part by a grant from the US-Israel Binational Science Foundation.
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References [ I] GE Thomas and W.J. M&h, Mol. Phys. 46 (1982) 951; W.J. Meath and EA. Power, J. Phys. B17 (1984) 763; Mol. Phys. 5 1 ( 1984) 585; B. Dick and G. Hohlneicher, J. Chem. Phys. 76 (1982) 5755. [ 21 W.M. McClain and R.A. Harris, in: Excited states, Vol.3, ed. E.C. Lim (Academic Pres, New York, 1977) pp. 2-56; B. Honig, J. Jortner and A. Szoke, J. Chem. Phys. 46 (1967) 2714. [3] W.L. Peticolas, J. Chem. Phys. 42 (1965) 4164; Ann. Rev. Phys. Chem. 18 (1967) 233. [ 41 N. Bloembergen, Nonlinear optics (Academic Press, New York, 1965) ch. 2. [5] Y.B. Band and B.M. Aizenbud, Chem. Phys. Letters 77 (1981) 49. [ 6) Y.B. Band and B.M. Aizenbud, Chem. Phys. Letters 79 ( 1981) 244. [ 71 S.J. Singer, S. Lee, K.F. Freed and Y.B. Band, J. Chem. Phys., to be published. [ 8 ] M. Goppert-Meyer, Ann. Physik (Leipzig) [ 51 9 ( 1931) 273. [ 91 E.T. Jaynes and F.W. Cummings, Proc. IEEE 5 1 (1963) 89; T.L. Knights and P.W. Milloni, Phys. Rept. 66 (1980) 22; C. Cohen-Tannoudji and S. Reynaud, J. Phys. Al0 ( 1977) 345; F.H. Mies and Y. Ben Aryeh, J. Chem. Phys. 74 (1981) 53. [ IO] F.H. Mies and Y.B. Band, to be published. [ 111 M.S. Child and H.C. Longuet-Higgins, Phil. Trans. Roy. Sot. (London) A254 (1961) 259. [ 121 B. Scharfand Y.B. Band, J. Chem. Phys. 79 (1983) 3175,3182. [ 131 B. ScharfandT.A. Miller, J. Chem. Phys. 84 (1986) 561. [ 141 B. Schatf, Chem. Phys. Letters 96 (1983) 89; 98 (1983) 81; 99 (1983) 350; 102 (1983) 184; 114 (1985) 291. [ 151 R.A. Kennedy, T.A. Miller and B. Scharf, J. Chem. Phys. 85 (1986) 1336.
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