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Permanent dipole moments and multi-photon resonances
Volume 108A, number 7
PHYSICS LETTERS
PERMANENT DIPOLE MOMENTS AND MULTI-PHOTON
15 April 1985
RESONANCES
M a r y A n n K M E T I C 1 a n d W d h a m J. M E A T H 1
Department of Chem,strv. Unwersm' of Western Ontarm. London. Canada Recmved 14 December 1984, rewsed manuscript recmved 11 February 1985, accepted for pubhcatmn 13 February 1985
The effects of non-zero dmgonal &pole matrix elements on multi-photon resonance profiles, for the mteractton of molecules with a smusoldal field, are &scussed by using an expressmn for the profiles derived m the rotating wave approxlmatmn The vah&ty of this result ts discussed with reference to exact two-level model calculatmns
The purpose o f this letter is to present two-level rotating wave approximations (RWA) that are useful in illustrating some o f the effects o f non-zero diagonal dipole matrix elements ("permanent dipole moments") on multip h o t o n resonance profiles. While such effects are implicitly included in any calculations on systems possessing permanent dipole moments, there are apparently only a few papers, for example refs. [ 1 - 4 ] , exphcltly concerned with this topic since most o f the detailed analysis o f multi-photon transitmns has been carried out for atonuc systems. Atomic units are used throughout what follows. The time-evolutmn o f a twodevel system is governed b y the matrix dlfferentml equation
id(Cl)=H(t)e=(HllH12](cl) dt
\H21H22]\c 2
c2
(1)
"
In the Schrodinger representation, after removing the traces o f the original energy and dipole matrices following ref. [3 ] , the time-dependent wavefunction is given b y q/(r, t ) = a 1(t)~b 1(r) + a2(t ) ~2(r), times a factor that does not contribute to the transition probabilities, where the (al(r) are the orthonormal time-independent wavefunctions for the stationary states having energy &j and e = a is the solution o f (1) with
(10)
H = ~1 [ A ~ - d E ( t ) ]
0
(01)
1 - pE(t) 1
0 "
Here A& = ~ 2 - & 1 > 0, d = It 22 - It 11, It = It 12, It ii = (q~iI It I~! ), d and ta are t he projections o f d and It along the direction o f polarization ~ o f the applied electric field E(t) = kE 0 s i n ( w t + 6) which has amplitude E 0, circular frequency w, and phase 6. In order to help locate the resonances in the transition probabilities it is useful to transform (1) into an interaction representation defined b y
a(1,2 ) = b(1,2 )
exp[(+,-)~i(A@
t-d
t f E(t')dt'
) .
0
'~ Research supported by a grant from the Natural Scmnces and Engineering Research Councd of Canada and through the award of a Province of Ontario Graduate Scholarship to MAK 1 Associated with the Centre for lnterdtsophnary Studies m Chemical Physics 340
0.375-9601/85/$ 03.30 © Elsevier Science Pubhshers B.V. (North-Holland Physxcs Pubhshmg Dwlslon)
Volume 108A, number 7
PHYSICS LETTERS
15 April 1985
The coefficients b satisfy (1) with H 11 = H22 = 0 and
I
HI2 =H~I = - p E ( t ) exp - i ( A ~ t - d ×
~ k=
_
/
E(t')dt'
t
= -~pE 0 exp[
Jk(dEO/w)exp(ik6){exp(16)exp[-i(A&-co-kco)t]
cosin
°I
+exp(-16)exp[-l(A~+co-kco)t]),
(2)
oo
where we have used exp(tz sin x)
k =_:_oo Sk(~)
exp(ikx)
and Jk(z) is a Bessel function of integer order k. The resonances occur whenever A & = (k + 1) co or A ~ = (k - 1) co and neglecting all other terms in (2), which correspond to off-resonance or counter-rotating terms, leads to a simple RWA approximation for H, namely H 11 = H22 = 0, 0 H12 = - ~1 (p.E)eeff exp {-i(A& -Nco)t - i[(dEO /co)sin 6 - N6 ] },
for the N-photon resonance (co ~ AS/N), where
(uE°)~ff = uE° [JN_ l (dEO /w) + JN+ l (dE° /w)] = uEO2~V(dEO/co)- l JN(dEO /co) ,
(3)
and to the following result for the steady state population of level 2 (corresponding to a 1(0) = 1, a2(0 ) = 0): 0N 2 r [(P~7 )eft[ ff~V = lim r - 1 f la 2 (t) l 2dr = r-• 0 2[(A& -NCO) 2 + [ (/~0)Nffl 2]
(4)
A generalization of (4), which includes the effects o f an apphed static electric field E0, can be obtained by combining our approach with that of ref. [3]; the derivation, which reqmres both a phase and long time average, is relatwely tedious and will be presented elsewhere. The result can be obtained from (4) by adding the term {(A& - - N w ) 2 - [(A& _ Nco) cos 20 - (pE0)eNff JN(dEO/CO) sm 20] 2)to the numerator and by replacing, in the resulting expression, A&,/a and d by, in the notation of ref. [3], 7 = [(A& -- aE0) 2 + 4 I/TtE012] 1/2, M =/.t cos 20 1 + i d sin 20 and D = d cos 20 - 2p sin 20 respectively, where cos 20 = (A& -- a E 0 ~ - 1, sin 20 = 2/2E07-1, and fi and d are the projections o f ~t and d along E 0. To simplify matters we take E 0 = 0 in what follows. As d ~ 0, (/zE0)Nff = tIEO6N, 1 since Jk(O) = 8k, 0 and (4) yields the well-known RWA result for a one-photon transition in the absence o f permanent dipole moments. Eq. (4), for arbitrary N, is a generalization of the standard result obtained b y replacing the usual coupling between atom (molecule) and field, namely btE 0, by an effective frequency dependent coupling term (3). At this level of approximation the effects of permanent dipoles arise from their interaction wath the applied field which causes a modulation of the usual coupling term by the multiphcative factor involving the Bessel function o f argument dEO/co = (P22 - / a l 1 ) EO/co in eq. (3); all such effects vanish if /'/22 =/dl 1When d 4= 0 at least one o f the states of the system has no definite parity and therefore both even and odd multi-photon transitions can take place, N = 1,2, 3, 4, ... in eqs. (3) and (4). The resonances occur at co = Ag/N. have a maxima o f 0.5, and oscillatory fringes with their zeros occurring at the zeros of JN(dE°]co) as a function o f co. For sufficiently narrow (main) resonances, where (dE°/co)- 1JN(dEO/co ) does not vary appreciably over the width o f the resonances, the frequency dependent full width at half maxima o f the resonances is given by (FWHM) N = 2[NI (pE0)eNfr t with co = A&/N. For fixed parameters of the problem this result suggests that the width of the 341
Volume 108A, number 7
PHYSICS LETTERS
15 April 1985
0 50-
d
0 25"
000-
-, ,-,_A
I
l
7
C
02~
0 O0
i
b
0 25
0 O0
i
a
025
0 O0 - O2
09 L01AE
16 0 2
09 L0/AE
16
Fig. 1. The steady state transition probability P2, as a function ~o/A~, for twoqevel model systems specified by/z = 1.0, d = 20.0, A& = 1.0, E ° = 0.5 (left) and ~t = - 0 . 5 0 7 2 , d = 2.0, A& = 3.706 X 1 0 - s , E ° = 5 X 10 -4 (right). In both eases ( a ) - ( e ) correspond to eq. (4) and (d) to an exact calculation [5].
342
Volume 108A, number 7
PHYSICS LETTERS
15 April 1985
mare resonances decrease as N increases, both through the factor N -1 and since JN(dEON/A,~) generally decreases as N increases. While this pre&ction will often be true, it is not rigorous, for example, if dEON/A~ is near a zero ofJN for N small relative to N large. Finally we pomt out our result is clearly not acceptable for small d when N = 3, 5, 7 .... since fiN given by (4) goes to zero as d ~ 0, a result valid only for even values o f N . Eq. (4) is apphcable for N = 1 as d -+ 0, yielding the usual lorentzlan resonance profile, with a frequency independent FWHM of 2 I/aE 0 I, associated with the standard RWA result. For N = 1 the effect o f a non-zero d is to introduce an asymmetry and oscillatory fringes into the resonance profde as a function o f 6o. The above discussion is illustrated by results for pN, N = 1,2 and 3, obtained from eq. (4) for the two-level model systems specified in the caption o f fig. 1. For comparison purposes the figure also contains the corresponding exact results for if2 obtained using Floquet techniques [5,6]. Exact two-level results for the choice o f parameters yielding in/aE0/A& = 6.84 and dEO/Ag = 161 can be found in the literature [2] and are apparently the first (explicit) example o f the effects, due to d ~ 0, discussed in this paper; this example is intermediate to those m fig. 1 but the agreement with eq. (4) is much more analogous to that illustrated in the left part o f fig. 1. The figure shows clearly that eq. (4), like all RWA type approximations, becomes more reliable as the coupling between the transition dipole and the applied field, here (/aE0)Nff, becomes small. Generally eq. (4) becomes more reliable as/aE ° decreases and as dE 0 increases for a fixed value o f A g and bearing in mind that the effects of neighbourmg states must be investxgated for many level problems. Fig. 1 (left) shows a typical example (/aE0/A& = 0.5, dEO/Ag = 10.0) where the RWA works very well whereas the right part o f fig. 1 (pE0/A& = 6.84, dE0/A g = 26.98) gaves and illustration o f when the RWA predicts the gross features o f the structure of the spectra but not important details like peak heights and a shift of the resonances to the low-frequency side o f the zero field resonance positions (co = A&/N);these shafts are just barely noticeable in the intermediate case calculation [2] referred to previously. The "near" resonance validity of the RWA result eq. (4) is clearly evident from the figure. The corrections to the RWA result can be mvestigated by using the expansion for H, In Fourier components, given by (2) to obtain a Floquet secular equation following Sharley [7] for the case d = 0. The development o f analytic correction terms is complicated and this, and a variety o f exact model calculations on two- and few-level systems, will be published in due course with the object o f elucidating the validity of eq. (4) in more detail and explaining details o f the effects o f non-zero permanent dipoles, for example the shaft to low frequency o f the resonances in fig. ld (right) with respect to the zero field positions relative to the shift to high frequency normally observed when d = 0. Our result eq. (4) corresponds to the zeroth plus first order terms of a complete Floquet perturbatlve analysis o f the problem. Finally we emphasize that the results o f this paper are valid for a non-rotating molecule; performing a rotational average [5] o f P 2 about the direction of the applied sinusoidal field removes the oscillatory fringes and narrows the resonances seen in the figure. The authors would like to thank Dr. R.A. Thuraisingham for interesting discussions.
References [1] [2] [3] [4] [5] [6] [7]
B. Dick and G. Hohlnelcher, J. Chem. Phys. 76 (1982) 5755. G.F. Thomas and W.J. Meath, Molec. Phys. 46 (1982) 743; 48 (1983) 649. W.J. Meath and E.A. Power, Molec. Phys. 51 (1984) 585. W.J. Meath and E.A. Power, J. Phys. BI7 (1984) 763. R.A. Thuralsingham and W.J. Meath, to be published. J.V. Moloney and W.J. Meath, Molec. Phys. 31 (1976) 1537. J.H. Shtrley, Phys. Rev. B138 (1965) 979.
343
PHYSICS LETTERS
PERMANENT DIPOLE MOMENTS AND MULTI-PHOTON
15 April 1985
RESONANCES
M a r y A n n K M E T I C 1 a n d W d h a m J. M E A T H 1
Department of Chem,strv. Unwersm' of Western Ontarm. London. Canada Recmved 14 December 1984, rewsed manuscript recmved 11 February 1985, accepted for pubhcatmn 13 February 1985
The effects of non-zero dmgonal &pole matrix elements on multi-photon resonance profiles, for the mteractton of molecules with a smusoldal field, are &scussed by using an expressmn for the profiles derived m the rotating wave approxlmatmn The vah&ty of this result ts discussed with reference to exact two-level model calculatmns
The purpose o f this letter is to present two-level rotating wave approximations (RWA) that are useful in illustrating some o f the effects o f non-zero diagonal dipole matrix elements ("permanent dipole moments") on multip h o t o n resonance profiles. While such effects are implicitly included in any calculations on systems possessing permanent dipole moments, there are apparently only a few papers, for example refs. [ 1 - 4 ] , exphcltly concerned with this topic since most o f the detailed analysis o f multi-photon transitmns has been carried out for atonuc systems. Atomic units are used throughout what follows. The time-evolutmn o f a twodevel system is governed b y the matrix dlfferentml equation
id(Cl)=H(t)e=(HllH12](cl) dt
\H21H22]\c 2
c2
(1)
"
In the Schrodinger representation, after removing the traces o f the original energy and dipole matrices following ref. [3 ] , the time-dependent wavefunction is given b y q/(r, t ) = a 1(t)~b 1(r) + a2(t ) ~2(r), times a factor that does not contribute to the transition probabilities, where the (al(r) are the orthonormal time-independent wavefunctions for the stationary states having energy &j and e = a is the solution o f (1) with
(10)
H = ~1 [ A ~ - d E ( t ) ]
0
(01)
1 - pE(t) 1
0 "
Here A& = ~ 2 - & 1 > 0, d = It 22 - It 11, It = It 12, It ii = (q~iI It I~! ), d and ta are t he projections o f d and It along the direction o f polarization ~ o f the applied electric field E(t) = kE 0 s i n ( w t + 6) which has amplitude E 0, circular frequency w, and phase 6. In order to help locate the resonances in the transition probabilities it is useful to transform (1) into an interaction representation defined b y
a(1,2 ) = b(1,2 )
exp[(+,-)~i(A@
t-d
t f E(t')dt'
) .
0
'~ Research supported by a grant from the Natural Scmnces and Engineering Research Councd of Canada and through the award of a Province of Ontario Graduate Scholarship to MAK 1 Associated with the Centre for lnterdtsophnary Studies m Chemical Physics 340
0.375-9601/85/$ 03.30 © Elsevier Science Pubhshers B.V. (North-Holland Physxcs Pubhshmg Dwlslon)
Volume 108A, number 7
PHYSICS LETTERS
15 April 1985
The coefficients b satisfy (1) with H 11 = H22 = 0 and
I
HI2 =H~I = - p E ( t ) exp - i ( A ~ t - d ×
~ k=
_
/
E(t')dt'
t
= -~pE 0 exp[
Jk(dEO/w)exp(ik6){exp(16)exp[-i(A&-co-kco)t]
cosin
°I
+exp(-16)exp[-l(A~+co-kco)t]),
(2)
oo
where we have used exp(tz sin x)
k =_:_oo Sk(~)
exp(ikx)
and Jk(z) is a Bessel function of integer order k. The resonances occur whenever A & = (k + 1) co or A ~ = (k - 1) co and neglecting all other terms in (2), which correspond to off-resonance or counter-rotating terms, leads to a simple RWA approximation for H, namely H 11 = H22 = 0, 0 H12 = - ~1 (p.E)eeff exp {-i(A& -Nco)t - i[(dEO /co)sin 6 - N6 ] },
for the N-photon resonance (co ~ AS/N), where
(uE°)~ff = uE° [JN_ l (dEO /w) + JN+ l (dE° /w)] = uEO2~V(dEO/co)- l JN(dEO /co) ,
(3)
and to the following result for the steady state population of level 2 (corresponding to a 1(0) = 1, a2(0 ) = 0): 0N 2 r [(P~7 )eft[ ff~V = lim r - 1 f la 2 (t) l 2dr = r-• 0 2[(A& -NCO) 2 + [ (/~0)Nffl 2]
(4)
A generalization of (4), which includes the effects o f an apphed static electric field E0, can be obtained by combining our approach with that of ref. [3]; the derivation, which reqmres both a phase and long time average, is relatwely tedious and will be presented elsewhere. The result can be obtained from (4) by adding the term {(A& - - N w ) 2 - [(A& _ Nco) cos 20 - (pE0)eNff JN(dEO/CO) sm 20] 2)to the numerator and by replacing, in the resulting expression, A&,/a and d by, in the notation of ref. [3], 7 = [(A& -- aE0) 2 + 4 I/TtE012] 1/2, M =/.t cos 20 1 + i d sin 20 and D = d cos 20 - 2p sin 20 respectively, where cos 20 = (A& -- a E 0 ~ - 1, sin 20 = 2/2E07-1, and fi and d are the projections o f ~t and d along E 0. To simplify matters we take E 0 = 0 in what follows. As d ~ 0, (/zE0)Nff = tIEO6N, 1 since Jk(O) = 8k, 0 and (4) yields the well-known RWA result for a one-photon transition in the absence o f permanent dipole moments. Eq. (4), for arbitrary N, is a generalization of the standard result obtained b y replacing the usual coupling between atom (molecule) and field, namely btE 0, by an effective frequency dependent coupling term (3). At this level of approximation the effects of permanent dipoles arise from their interaction wath the applied field which causes a modulation of the usual coupling term by the multiphcative factor involving the Bessel function o f argument dEO/co = (P22 - / a l 1 ) EO/co in eq. (3); all such effects vanish if /'/22 =/dl 1When d 4= 0 at least one o f the states of the system has no definite parity and therefore both even and odd multi-photon transitions can take place, N = 1,2, 3, 4, ... in eqs. (3) and (4). The resonances occur at co = Ag/N. have a maxima o f 0.5, and oscillatory fringes with their zeros occurring at the zeros of JN(dE°]co) as a function o f co. For sufficiently narrow (main) resonances, where (dE°/co)- 1JN(dEO/co ) does not vary appreciably over the width o f the resonances, the frequency dependent full width at half maxima o f the resonances is given by (FWHM) N = 2[NI (pE0)eNfr t with co = A&/N. For fixed parameters of the problem this result suggests that the width of the 341
Volume 108A, number 7
PHYSICS LETTERS
15 April 1985
0 50-
d
0 25"
000-
-, ,-,_A
I
l
7
C
02~
0 O0
i
b
0 25
0 O0
i
a
025
0 O0 - O2
09 L01AE
16 0 2
09 L0/AE
16
Fig. 1. The steady state transition probability P2, as a function ~o/A~, for twoqevel model systems specified by/z = 1.0, d = 20.0, A& = 1.0, E ° = 0.5 (left) and ~t = - 0 . 5 0 7 2 , d = 2.0, A& = 3.706 X 1 0 - s , E ° = 5 X 10 -4 (right). In both eases ( a ) - ( e ) correspond to eq. (4) and (d) to an exact calculation [5].
342
Volume 108A, number 7
PHYSICS LETTERS
15 April 1985
mare resonances decrease as N increases, both through the factor N -1 and since JN(dEON/A,~) generally decreases as N increases. While this pre&ction will often be true, it is not rigorous, for example, if dEON/A~ is near a zero ofJN for N small relative to N large. Finally we pomt out our result is clearly not acceptable for small d when N = 3, 5, 7 .... since fiN given by (4) goes to zero as d ~ 0, a result valid only for even values o f N . Eq. (4) is apphcable for N = 1 as d -+ 0, yielding the usual lorentzlan resonance profile, with a frequency independent FWHM of 2 I/aE 0 I, associated with the standard RWA result. For N = 1 the effect o f a non-zero d is to introduce an asymmetry and oscillatory fringes into the resonance profde as a function o f 6o. The above discussion is illustrated by results for pN, N = 1,2 and 3, obtained from eq. (4) for the two-level model systems specified in the caption o f fig. 1. For comparison purposes the figure also contains the corresponding exact results for if2 obtained using Floquet techniques [5,6]. Exact two-level results for the choice o f parameters yielding in/aE0/A& = 6.84 and dEO/Ag = 161 can be found in the literature [2] and are apparently the first (explicit) example o f the effects, due to d ~ 0, discussed in this paper; this example is intermediate to those m fig. 1 but the agreement with eq. (4) is much more analogous to that illustrated in the left part o f fig. 1. The figure shows clearly that eq. (4), like all RWA type approximations, becomes more reliable as the coupling between the transition dipole and the applied field, here (/aE0)Nff, becomes small. Generally eq. (4) becomes more reliable as/aE ° decreases and as dE 0 increases for a fixed value o f A g and bearing in mind that the effects of neighbourmg states must be investxgated for many level problems. Fig. 1 (left) shows a typical example (/aE0/A& = 0.5, dEO/Ag = 10.0) where the RWA works very well whereas the right part o f fig. 1 (pE0/A& = 6.84, dE0/A g = 26.98) gaves and illustration o f when the RWA predicts the gross features o f the structure of the spectra but not important details like peak heights and a shift of the resonances to the low-frequency side o f the zero field resonance positions (co = A&/N);these shafts are just barely noticeable in the intermediate case calculation [2] referred to previously. The "near" resonance validity of the RWA result eq. (4) is clearly evident from the figure. The corrections to the RWA result can be mvestigated by using the expansion for H, In Fourier components, given by (2) to obtain a Floquet secular equation following Sharley [7] for the case d = 0. The development o f analytic correction terms is complicated and this, and a variety o f exact model calculations on two- and few-level systems, will be published in due course with the object o f elucidating the validity of eq. (4) in more detail and explaining details o f the effects o f non-zero permanent dipoles, for example the shaft to low frequency o f the resonances in fig. ld (right) with respect to the zero field positions relative to the shift to high frequency normally observed when d = 0. Our result eq. (4) corresponds to the zeroth plus first order terms of a complete Floquet perturbatlve analysis o f the problem. Finally we emphasize that the results o f this paper are valid for a non-rotating molecule; performing a rotational average [5] o f P 2 about the direction of the applied sinusoidal field removes the oscillatory fringes and narrows the resonances seen in the figure. The authors would like to thank Dr. R.A. Thuraisingham for interesting discussions.
References [1] [2] [3] [4] [5] [6] [7]
B. Dick and G. Hohlnelcher, J. Chem. Phys. 76 (1982) 5755. G.F. Thomas and W.J. Meath, Molec. Phys. 46 (1982) 743; 48 (1983) 649. W.J. Meath and E.A. Power, Molec. Phys. 51 (1984) 585. W.J. Meath and E.A. Power, J. Phys. BI7 (1984) 763. R.A. Thuralsingham and W.J. Meath, to be published. J.V. Moloney and W.J. Meath, Molec. Phys. 31 (1976) 1537. J.H. Shtrley, Phys. Rev. B138 (1965) 979.
343