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Preparation and decay of excited molecular states: the influence of dephasing relaxation and pulse fall time
Volume 58, number2
PREPARATION THE INFLUENCE
CHEhfICAL
PHYSICS
LEITERS
AND DECAY OF EXCITED MOLECULAR OF DEPHASING
RELAXATION
15 Se;-;.xn3er 1978
STATES:
AND PULSE FALL TIME
Paolo GRIGOLINI Ldmratorio di Clhimica QuantisticG ed Energetica Molecokzre del C&R., 566100 P&z, Itaiy Received 24 May 1978
Revised manuscript received 13 June 1978
It is shown that both a tinite excitation pulse fall time and a transverse relaxation process allow a non-markovian radiationless process to exhibit a rigorous exponential behavior, whereas a long duration square pulse in the absence of dephasing relaxation does not succeed in fully elimiiting any memory feature_
1. Introduction The problem of preparation and decay of unstable states has widely been discussed in several fields of research [I-6] _As far as radiationless decay in molecules is concerned, great emphasis, in the case of minimum uncertainty irradiation, has been placed on the role of the pulse time duration in determining the decay characteristics [2--51. Furthermore, a natura! time limit to the possibility of regarding molecules as isolated systems is usually included in current approaches [6] through a damping contribution to the effective hamiltonian, which gives rise to depopulation effects. No attention has been devoted, to our knowledge, to studying interactions of adiabatic kind with the external world and their effects on intramolecular rehrxation processes of non-markovian nature. In our previous paper on preparation and decay of excited moiecular states [5] we have obtained results which qualitatively agree with those of ref- [3] _ However, a rigorous exponential decay could not be achieved by increasing the excitation pulse time duration. In fact, the non-markovian decays after long duration pulses exhibited a sort of stepwise form, even in the case where the irradiation width was very sharp with respect to the profile of the interaction density function [S] _ In this note we will show that the exact shape of the pulse, and especially its outgoing part, can give a significant contribution to eliminating the above mentioned stepwise form. It will also be shown that a further significant contribution can be provided by adiabatic interactions with external “thermal baths”. In order to obtain the latter result we shah generalize the theory of ref. [5] . Such an improved theory wii be used also in the companion note [7] _
2. Theory
We assume the hamiltonian of the system under study to be written as
(1) where 3$‘) is the hamiltonian associated with the vector space of interest. For simplicity we also assume that the space of interest involves only a “ground” state b$, with energy 4, and an excited state le), with energy e,. These two states are coupled between themselves by an external electromagneric excitation. Such an interaction is express185
CHEMICAL.
Volume 58. number2
PHYSICS
JITTJZRS
15
September 1978
ed as follows
.X$:&t> = 2(If?) <9l f lg> (eI)F(t)Hcos
wt -
The intramolecular “the,mal bath” hamikonian 3@
is given by
where X,,, is the interaction between a dense set of states, {IL&), and the suitable state continuum one usually introduces [6] in order to take into account the existence of a natural time limit, l/I’, to isolated molecular behavior. Such a dissipative mec&mism could be described in a more realistic way by emphasizing its longitudinal nature 181. However, in this paper we shall neglect such a detailed analysis and shaU focus our attention on the interactions with external world (XL”) of adiabatic type, X$2_ Following ref. [8], we perform the markovian assumption on the interrtction X(l) SB and obtain the well-known Redfield equation [P] (we recall AXB = AB - BA) (4) where
=_(eI~lg)/T~
(5)
and, for any state Im>,
The main difference with respect to ref_ [S] is the presence of a transverse relaxation process affecting also the mat& elements (elp fmX One could neglect this dephasing process in the case, for example, where the states Im) are undergoing the same enerw fluctuations as the ones affecting the state le>, and the state 18) is not affected by any energy fluctuation. In the case where le) were the only state undergoing fluctuation, the matrix elements (e Ip Im> would exhibit the same relaxation process as (e lp IgX For simplicity, in the present note and in the subsequent one [7] we shall use the latter approximation instead of introducing a lot of independent lifetime parameters. in the presence of a traa.sverSe relaxation process involviig also the matrix elements
A(O)= xg”)+ [
cm
Em
lm)Onl
1” +iR=x(~)x
+*
J
as the unperturbed part of the superoperator acting on the r.hs. of eq_ (4) and A(,‘) = (J&z f X!&)’ as its perturbation part. Then we perform the relative transformation to the interaction frame ofreference. After defining the projection operator P in such a way that Pp=le~~elple~~eI~Ie~~elpIg~~gl+Ig~~glple~~eI~~~(glpIg~dol. we can apply the Zwanzig approach
[IO] to eq. (4) written in the interaction picture. Then, we obtain (5 =pi3)
t iaC//ai = 1 @t, r) is (r) dr + (%&t(r))” 0
186
5 ,
(7)
CHEMICAL PHYSICS LETTERS
Volume 58, number2
15 September1978
where z = exp(iAf%)o (G&t
5(f,
_
(8)
jjx= exp(in<‘Q) T)=
X(l)(t) exp_
(3c~~(r))x
exp(-il\‘o’t)
-ij &‘(I E T
-P);i”)(t’)]
(9)
9
(1 -P)W(r)
and P)(T)
I\<‘) exp (-iA(*
= exp (in”%)
_
(10
using the same approach as the one of ref. [5] it is possible to show that 70, r) can be expressed as a function of both electromagnetic contributions, which depend only on the system of interest, and intramo!ecular contributions which in turn depend on the foilowing second-order correlation function
@*‘(t ) =
= e-
r2
exp (-LX (Ojx t)eLRt3&,,
le) _
(12)
= T2 for any m aliows us to write (r2 = 1/T2)
r #g&z, mtIa(f)
(13)
f
where & is just the second-order correlation function that one would obtain if the coupling X&i vanished. In ref_ [S] z?have shown that @jt2) rtra(t) can usually be expressed as a sum of complex exponentids. In the most simple case, for example, where C@::&(t) = $g2y2 e-yt
,
(14)
the presence of the transverse relaxation compels us to replace the memory parameter g with the effective one g’ = ,gy/(r f rz). AS usually done [5,8], in this case VJ~ can introduce the “reduced” molecular system involving only the three states le>, 18) and IM) driven by the hamiltonian Xefl=
le>(ce -iyr.&(eI+IM)(e-i~)W+Ig)eg(gl
f u(IM)
<8l+ lg>
cos wt
(1%
(in this note we shall assume that TV, the radiative damping, is negligible with respect tc the other physical parameters involved by the “reduced” system) and by a transverse relaxation process which compels all the off-diagonal matrix elements concerning the state le) to decay in a time T2_
3. Applications For the sake of simplicity we shah investigate separately the effects of a finite fall of the pulse and those of a transverse relaxation process. In the case where l/T2 = 0 the time evolution of the system under study can be obtained by solving the SchrZjdinger equation related to the hamiltonian of eq. (15). In the case where the exciting irradiation field is fairly weak we can regard the last term on the rhs. of eq. (15), X$ii(t>, as a smah perturbation. The Schrikiinger equation in the interaction picture [ZC$$= XeE - XT,!&] ia
@)/at= Z&(t)@)
where %TeXt= exp(i
(16)
,
(*If )3C,,, exp(-iX(*)t)
and 13) = exp(iJC’*)t)l
W, has the formal solution
187
VchIme
58, number
cxE%f ICAE
2
PfiYS1cs
-Rs
15 September1978
In the case where the electromagnetic fiefd is very weak and the system’s initial condition is expressed by f%?(O))= ig), we have (eI;i;(r)> = -i jk4&&~kJdt’*
(18)
0 The analyticaIexpression of Ze, only requiresthe knowledge of the eigenstatesand eigenvaiueso~X(~), which can be obtained sim$y by diagonaMng a two-dimensional matrix. As f&r as the puJ.seshape is concerned, we follow ret fZ I] and use the realisticquasi-rectangdar pulse of the fOITI2 F(t)
= expb3
0
= I
=
exp(- 7zt)
@
(W
(0Gt
WI
(C>S).
(W)
Then we obtain
where 1=I,
respectively;AC+ = tee -
(O=st
=
f frr + i(Ez -t Aw,)]
. exp [i(E, -r
f Aw,)T]
E =J.=-(G+i)r/Z, (el;rl>= f(G-i)/2G11/‘, 188
- f (20
i(E, + AU,)
The eigenvaluesand eigenstatesare defined by the follow@ En2 =(G-i)r/2, -
(213
relations (Q = erM>
(22) (=I
Volume 58, number 2
C34!ZMIW
PMYSICSLlZiTERS
Fig. 1. Time dependenceof excited state popukttion.The rise and f& times of excitationpubes areprovidedby the fat lowing time constantsy1 = r2 = 27 <---); yt = 7z = 10 y (--). The p!ateaulengthsare: T = 1 (curves1 and 2); T= 3 (curve3); T= 10 (curves4 and 5). Time is expressedin units of l/r_ awl = e/2.
15 Septeniber1978
Fig_2. Time dependenceof excited statepopulation. The time durationof the rectangular excitationp&se is T= I. Tile trmsversal relaxationratesare: r2 = 0 (-), I-2 = -y (---), r2 =: 10 y (-_-_-)_ Time is expressedin units of I/r. Awl = 0.
where G =
(-g2- #f’_
(24)
The pufse shape used by Berg et al. in their theory of resonance scattering [IZ] can be expressed by using eqs. (13) provided that Tis assumed zero. If we perform also the markovian assumption (g < l), the contribution of the state 13r2) to the summation of eq. (17) vanishes, the real part of E_ vanishes sd its imaginary part is just the damping parameter denoted by y in ref. [ 12]_ Then the square modulus of
189
Voiume 58, number 2
CHE_MJCAL PHYSK!s
irEmE=
-15 September 1978
References [I J L. Fonde, in: Aoceedings of the XJII Winter School of Theoretical Physics, Karpaa, i2] k Ziy and W. Rhodes, J. Chem- whys_ 65 (1976) 4895, and references therein. [3] W- Rhodes, Chem- Phys. 22 (1977) 95_ [4] G-W- Robinson and CA- Lan&off, Chem Phys. 5 (1974) 1. ISI P. GrigoIini and A- bmi. Chem. Phys_ 30 (1978) 61. [61 K-F. Freed, Topics Current C&em. 31 (1972) 105;;. Chem. pfiys. 52 (1970) 1345 _ [7] P. Gr&olini, Chem. Ph;rr Letrers 58 (1978) 191_ [8] P- Grigolini and A- Lami. (Ihem_ Phys. Letters 5.5 (1978) 152. 19) A-G- Redfield, in: Advances in magnetic resonance, VoL 1, ed. JS. [ 101 R-W- Zwan@. in: Lectures in tlreoretical physics. Vol. 3. eds. W-E. York. 1961) p_ 106; 0. Phtz, in: EJectron spin relaxation in Jiquids, eds. L.T. hfuus and [Ill A-D. WiIson and H. Friedmann, Chem_ Phys. 23 (1977) 105. [ 121 J-0. Berg. CA_ Langhoff and G-W. Robinson, Chem. Phys_ Letters
190
Poland (1976), and references therein.
Waugh (Academic Press, New York, 1965) p. 1. Brittin, B-W_ Downs and 1. Downs (Interscience, New P.W. Atkins (Plenum Press, New York, 1972) p. 89. 29 (1974) 305_
PREPARATION THE INFLUENCE
CHEhfICAL
PHYSICS
LEITERS
AND DECAY OF EXCITED MOLECULAR OF DEPHASING
RELAXATION
15 Se;-;.xn3er 1978
STATES:
AND PULSE FALL TIME
Paolo GRIGOLINI Ldmratorio di Clhimica QuantisticG ed Energetica Molecokzre del C&R., 566100 P&z, Itaiy Received 24 May 1978
Revised manuscript received 13 June 1978
It is shown that both a tinite excitation pulse fall time and a transverse relaxation process allow a non-markovian radiationless process to exhibit a rigorous exponential behavior, whereas a long duration square pulse in the absence of dephasing relaxation does not succeed in fully elimiiting any memory feature_
1. Introduction The problem of preparation and decay of unstable states has widely been discussed in several fields of research [I-6] _As far as radiationless decay in molecules is concerned, great emphasis, in the case of minimum uncertainty irradiation, has been placed on the role of the pulse time duration in determining the decay characteristics [2--51. Furthermore, a natura! time limit to the possibility of regarding molecules as isolated systems is usually included in current approaches [6] through a damping contribution to the effective hamiltonian, which gives rise to depopulation effects. No attention has been devoted, to our knowledge, to studying interactions of adiabatic kind with the external world and their effects on intramolecular rehrxation processes of non-markovian nature. In our previous paper on preparation and decay of excited moiecular states [5] we have obtained results which qualitatively agree with those of ref- [3] _ However, a rigorous exponential decay could not be achieved by increasing the excitation pulse time duration. In fact, the non-markovian decays after long duration pulses exhibited a sort of stepwise form, even in the case where the irradiation width was very sharp with respect to the profile of the interaction density function [S] _ In this note we will show that the exact shape of the pulse, and especially its outgoing part, can give a significant contribution to eliminating the above mentioned stepwise form. It will also be shown that a further significant contribution can be provided by adiabatic interactions with external “thermal baths”. In order to obtain the latter result we shah generalize the theory of ref. [5] . Such an improved theory wii be used also in the companion note [7] _
2. Theory
We assume the hamiltonian of the system under study to be written as
(1) where 3$‘) is the hamiltonian associated with the vector space of interest. For simplicity we also assume that the space of interest involves only a “ground” state b$, with energy 4, and an excited state le), with energy e,. These two states are coupled between themselves by an external electromagneric excitation. Such an interaction is express185
CHEMICAL.
Volume 58. number2
PHYSICS
JITTJZRS
15
September 1978
ed as follows
.X$:&t> = 2(If?) <9l f lg> (eI)F(t)Hcos
wt -
The intramolecular “the,mal bath” hamikonian 3@
is given by
where X,,, is the interaction between a dense set of states, {IL&), and the suitable state continuum one usually introduces [6] in order to take into account the existence of a natural time limit, l/I’, to isolated molecular behavior. Such a dissipative mec&mism could be described in a more realistic way by emphasizing its longitudinal nature 181. However, in this paper we shall neglect such a detailed analysis and shaU focus our attention on the interactions with external world (XL”) of adiabatic type, X$2_ Following ref. [8], we perform the markovian assumption on the interrtction X(l) SB and obtain the well-known Redfield equation [P] (we recall AXB = AB - BA) (4) where
=_(eI~lg)/T~
(5)
and, for any state Im>,
The main difference with respect to ref_ [S] is the presence of a transverse relaxation process affecting also the mat& elements (elp fmX One could neglect this dephasing process in the case, for example, where the states Im) are undergoing the same enerw fluctuations as the ones affecting the state le>, and the state 18) is not affected by any energy fluctuation. In the case where le) were the only state undergoing fluctuation, the matrix elements (e Ip Im> would exhibit the same relaxation process as (e lp IgX For simplicity, in the present note and in the subsequent one [7] we shall use the latter approximation instead of introducing a lot of independent lifetime parameters. in the presence of a traa.sverSe relaxation process involviig also the matrix elements
A(O)= xg”)+ [
cm
Em
lm)Onl
1” +iR=x(~)x
+*
J
as the unperturbed part of the superoperator acting on the r.hs. of eq_ (4) and A(,‘) = (J&z f X!&)’ as its perturbation part. Then we perform the relative transformation to the interaction frame ofreference. After defining the projection operator P in such a way that Pp=le~~elple~~eI~Ie~~elpIg~~gl+Ig~~glple~~eI~~~(glpIg~dol. we can apply the Zwanzig approach
[IO] to eq. (4) written in the interaction picture. Then, we obtain (5 =pi3)
t iaC//ai = 1 @t, r) is (r) dr + (%&t(r))” 0
186
5 ,
(7)
CHEMICAL PHYSICS LETTERS
Volume 58, number2
15 September1978
where z = exp(iAf%)o (G&t
5(f,
_
(8)
jjx= exp(in<‘Q) T)=
X(l)(t) exp_
(3c~~(r))x
exp(-il\‘o’t)
-ij &‘(I E T
-P);i”)(t’)]
(9)
9
(1 -P)W(r)
and P)(T)
I\<‘) exp (-iA(*
= exp (in”%)
_
(10
using the same approach as the one of ref. [5] it is possible to show that 70, r) can be expressed as a function of both electromagnetic contributions, which depend only on the system of interest, and intramo!ecular contributions which in turn depend on the foilowing second-order correlation function
@*‘(t ) =
= e-
r2
exp (-LX (Ojx t)eLRt3&,,
le) _
(12)
= T2 for any m aliows us to write (r2 = 1/T2)
r #g&z, mtIa(f)
(13)
f
where & is just the second-order correlation function that one would obtain if the coupling X&i vanished. In ref_ [S] z?have shown that @jt2) rtra(t) can usually be expressed as a sum of complex exponentids. In the most simple case, for example, where C@::&(t) = $g2y2 e-yt
,
(14)
the presence of the transverse relaxation compels us to replace the memory parameter g with the effective one g’ = ,gy/(r f rz). AS usually done [5,8], in this case VJ~ can introduce the “reduced” molecular system involving only the three states le>, 18) and IM) driven by the hamiltonian Xefl=
le>(ce -iyr.&(eI+IM)(e-i~)W+Ig)eg(gl
f u(IM)
<8l+ lg>
cos wt
(1%
(in this note we shall assume that TV, the radiative damping, is negligible with respect tc the other physical parameters involved by the “reduced” system) and by a transverse relaxation process which compels all the off-diagonal matrix elements concerning the state le) to decay in a time T2_
3. Applications For the sake of simplicity we shah investigate separately the effects of a finite fall of the pulse and those of a transverse relaxation process. In the case where l/T2 = 0 the time evolution of the system under study can be obtained by solving the SchrZjdinger equation related to the hamiltonian of eq. (15). In the case where the exciting irradiation field is fairly weak we can regard the last term on the rhs. of eq. (15), X$ii(t>, as a smah perturbation. The Schrikiinger equation in the interaction picture [ZC$$= XeE - XT,!&] ia
@)/at= Z&(t)@)
where %TeXt= exp(i
(16)
,
(*If )3C,,, exp(-iX(*)t)
and 13) = exp(iJC’*)t)l
W, has the formal solution
187
VchIme
58, number
cxE%f ICAE
2
PfiYS1cs
-Rs
15 September1978
In the case where the electromagnetic fiefd is very weak and the system’s initial condition is expressed by f%?(O))= ig), we have (eI;i;(r)> = -i jk4&&~kJdt’*
(18)
0 The analyticaIexpression of Ze, only requiresthe knowledge of the eigenstatesand eigenvaiueso~X(~), which can be obtained sim$y by diagonaMng a two-dimensional matrix. As f&r as the puJ.seshape is concerned, we follow ret fZ I] and use the realisticquasi-rectangdar pulse of the fOITI2 F(t)
= expb3
0
= I
=
exp(- 7zt)
@
(W
(0Gt
WI
(C>S).
(W)
Then we obtain
where 1=I,
respectively;AC+ = tee -
(O=st
=
f frr + i(Ez -t Aw,)]
. exp [i(E, -r
f Aw,)T]
E =J.=-(G+i)r/Z, (el;rl>= f(G-i)/2G11/‘, 188
- f (20
i(E, + AU,)
The eigenvaluesand eigenstatesare defined by the follow@ En2 =(G-i)r/2, -
(213
relations (Q = erM>
(22) (=I
Volume 58, number 2
C34!ZMIW
PMYSICSLlZiTERS
Fig. 1. Time dependenceof excited state popukttion.The rise and f& times of excitationpubes areprovidedby the fat lowing time constantsy1 = r2 = 27 <---); yt = 7z = 10 y (--). The p!ateaulengthsare: T = 1 (curves1 and 2); T= 3 (curve3); T= 10 (curves4 and 5). Time is expressedin units of l/r_ awl = e/2.
15 Septeniber1978
Fig_2. Time dependenceof excited statepopulation. The time durationof the rectangular excitationp&se is T= I. Tile trmsversal relaxationratesare: r2 = 0 (-), I-2 = -y (---), r2 =: 10 y (-_-_-)_ Time is expressedin units of I/r. Awl = 0.
where G =
(-g2- #f’_
(24)
The pufse shape used by Berg et al. in their theory of resonance scattering [IZ] can be expressed by using eqs. (13) provided that Tis assumed zero. If we perform also the markovian assumption (g < l), the contribution of the state 13r2) to the summation of eq. (17) vanishes, the real part of E_ vanishes sd its imaginary part is just the damping parameter denoted by y in ref. [ 12]_ Then the square modulus of
189
Voiume 58, number 2
CHE_MJCAL PHYSK!s
irEmE=
-15 September 1978
References [I J L. Fonde, in: Aoceedings of the XJII Winter School of Theoretical Physics, Karpaa, i2] k Ziy and W. Rhodes, J. Chem- whys_ 65 (1976) 4895, and references therein. [3] W- Rhodes, Chem- Phys. 22 (1977) 95_ [4] G-W- Robinson and CA- Lan&off, Chem Phys. 5 (1974) 1. ISI P. GrigoIini and A- bmi. Chem. Phys_ 30 (1978) 61. [61 K-F. Freed, Topics Current C&em. 31 (1972) 105;;. Chem. pfiys. 52 (1970) 1345 _ [7] P. Gr&olini, Chem. Ph;rr Letrers 58 (1978) 191_ [8] P- Grigolini and A- Lami. (Ihem_ Phys. Letters 5.5 (1978) 152. 19) A-G- Redfield, in: Advances in magnetic resonance, VoL 1, ed. JS. [ 101 R-W- Zwan@. in: Lectures in tlreoretical physics. Vol. 3. eds. W-E. York. 1961) p_ 106; 0. Phtz, in: EJectron spin relaxation in Jiquids, eds. L.T. hfuus and [Ill A-D. WiIson and H. Friedmann, Chem_ Phys. 23 (1977) 105. [ 121 J-0. Berg. CA_ Langhoff and G-W. Robinson, Chem. Phys_ Letters
190
Poland (1976), and references therein.
Waugh (Academic Press, New York, 1965) p. 1. Brittin, B-W_ Downs and 1. Downs (Interscience, New P.W. Atkins (Plenum Press, New York, 1972) p. 89. 29 (1974) 305_