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Theory of time-resolved level anticrossing experiment
ChemicalPhysics 52 (1980) 363-379 ©North-Holland Publishing Company
THEORY OF TIME-RESOLVED LEVEL ANTICROSSING EXPERIMENT R. HABERKORN, H.L. SELZLE, W. DIETZ, S.H. LIN * and E.W. SCHLAG Institute for Physical and Theoretical Chemistry, Technical University of Munich, 8046 Garching, West Germany Received 24 January, 1980; in final form 28 May, 1980
The purpose of the present paper is to present a general treatment of steady state and time-resolved level-anticrossing (LAC) experiments using the generalized master equation approach in which both radiative and nonradiative decays and dephasing processes have been considered. Several models are treated to demonstrate the theoretical method. The theory is applicable to both atoms and molecules. It is shown that under appropriate conditions, as derived in this paper, quantum beats can be observed in the time-resolved LAC experiment. Numerical calculations have been performed to show the time-dependent behavior (build-up and decay) of the time-resolved LAC experiment. These quantum beats are then a direct measurement of the microscopic coupling parameters in intersystem crossing, etc. It will be shown that combining the steady state and the time-resolved LAC experiments one can determine not only the microscopic coupling parameters but also the relaxation and/or the dephasing rate constants. Hence the particular virtues of time dependent LAC experiments are seen in these model calculations.
1 Introduction The level anticrossing (LAC) phenomenon is observed whenever two levels, say Im) and I n ) w i t h unperturbed energies E m and E n of an atom or a molecule are brought into coincidence by applying a magnetic field, provided that there exists some interaction H'mn = (m I/~' In) between Im } and In). Near the crossing point, the two eigenvalues of the total hamiltonian repel one another and the associated two eigenfunctions are mixtures o f Im } and In }. This produces an equilization of the populations of Im) and In } at the avoided crossing point if some steady state excitation mechanism produces a difference between the populations o f Im} and In }. This resonant-like variation of population can be observed by monitoring separately the intensity and/or polarization o f light emitted by Im} and In}. The i n t e r a c t i o n / 1 ' could be due to an electric field interaction, a fine or hyperfine interaction etc. Level anticrossing was observed for the first time on the sublevels of the 2 2 p manifold of Li; the coupling in this case was due to the hyperfine interaction [ 1 j. Since then a great number of other LAC experiments have been carried out for atoms and molecules [ 2 - 7 ] ~. However, all o f LAC experiments so far have been concerned 0nly with the measurement of steady state behavior. The purpose o f the present paper is to present a theoretical approach which can be applied to study not only the steady state behavior but also the dynamic behavior (i.e. time evolution) of LAC phenomena. To demonstrate the theoretical method, we consider the pseudo two-level system and pseudo four-level systems corresponding to LAC between two doublet states, and between a singlet and a triplet. Numerical calculations are performed to illustrate the effect of various parameters (e.g. energy level spacing, coupling constant H'mn, decay constants, dephasing constants, etc.) on the dynamic and steady state behavior o f the LAC phenomenon. It should be noted that although in this paper we deal only with LAC, clearly the present theoretical approach can be modified and applied to other related experiments like level crossing, double resonance, MOMRIE etc. * Permanent address: Department of Chemistry, Arizona State University, Tempe, Arizona 85281, USA. For a recent review, see ref. [ 21.
364
R. Haberkorn et al. / Theory o f time-resolved level anticrossing experiment
2. General considerations It has been shown that the time dependence of the density matrix of the system embedded in a heat bath is determined by the Liouville equation of motion [8,9] dr)/at = -i/~o~5 - i£'15- [ ' ~ ,
(2.1)
where Lo represents the Liouville operator from which the basis set is chosen, L' represents the interaction Liouville operator that causes the level anticrossing and I" represents the damping operator. In terms of matrix elements of ~, we find, for example dPnn_ dt
(2.2)
i ~ ( H , n n , P n , n - P n n ' H n ' n ) - ~ Pnn:n'n'Pn'n' h n' n'
and dpmn _ dt
(i~mn + Pmn'mn)Pmn - ~ "
'
(2.3)
(H~n'Pn', - Pmn'H~'n) .
These are usually called the master equations. Here Hnn' represents the interaction of the levels, which is responsible for the anticrossing, P describes the damping of the system and boron is the energy difference between the levels without interaction. For convenience of discussion, the properties of I~ are summarized in the following [8] !
Pnn:nn = --
(2.4)
rn'n':nn ,
f n
where Fn~nt
nn
27r ~ h %
-
~ p(b)(O)nbnb[(nno[i21,RNin,n,b)l= 6 ( E n ' n b _ nb
~""b)'
(2.5)
~(b)(0) represents the initial distribution of the density matrix of the heat bath and HRN denotes the interaction between the system and the heat bath (including both radiative and nonradiative intramolecular and intermolecular interactions), and Pmn :ran is the dephasing rate constant d) F m n :mn = ]1 ( F r o m : r a m + P n n : n n ) + P (ran:ran '
(2.6)
where P ran:ran (d) represents the so-called pure dephasing rate constant (d)
Pmn:mn
rr ~ = -h
~
nb
p(b)(O)n,bnb([(nnblICl,RNinn,b)l -- [(rnnb["HRNlmn'b)D ' 2 8(En b - En'b)
n'b
(2.7)
As can be seen from eq. (2.5), - P n ' n ':nn represents the rate constant for the transition n -+ n'. In particular, for the three-level system (a, m and n) eqs. (2.2) and (2.3) become dPnn/dt =
-
(i/h)[(H'naPan - PnaHan) ' + (HnmPmn ' - PnmHmn)] '
(2.8)
-- ['nn :nnPnn -- P n n : m m P m m -- Pnn :aaPaa , t
t
dPmn/dt = - (it°ran + Pmn :ran) Pmn -- (i/h)[(HmaPan - PmaHa,) + Hmn(Pnn - Pmm)] ,
(2.9)
respectively, where r
(.Omn
=
OOmn +
h--
I
t
(Hmm
-
t H..)
(2.10)
R. Haberkorn et al. / Theory o f time-resolved level anticrossing experiment
365
and for example Fnn:nn
= -Faa
(2.11)
: n n - - F r o m :nn •
The matrix elements of 1:" (or rate constants) are schematically shown in the following: n
~-Fnn:mm~:P_m__m!_nn__~-
7
m
I -Paa :nn
- P n n :aa i
It should be emphasized that P is well defined (i.e., can be calculated microscopically) and includes both radiative and nonradiative processes; for nonradiative processes, both intramolecular and intermolecular (if they exist) processes are included.
3. Three-level model We first consider the simple three-level system H m a
Here a represents the ground state and rn and n are the states in which the anti-level crossing (or interference) takes place. If we assume that in eqs. (2.8) and (2.9) klan = 0 ;
Ham
= O,
(3.1)
which means that there is no interaction of n and m with the ground state a, respectively, and eliminate Paa by using the relation (3.2)
Paa + Prom + Pnn = 1
we obtain the pseudo two-level problem dpmm/dt
= -
( 1" / h ) ( g m n' P n m
-
PmnHnm)'
- - F r 'o m : m m P m m
-
rmm'
:nnPnn
-
Pmm
t
dPnn/dt
= -
dPmn/dt
' -- PnmHmn)
(i/h)(H'nmPmn
= -iCOmnPm
n +
(i/h)
' Onn - P n n '. m m P m m - Fnn:aa Fnn:nn
-
H'mn(Pmm
-
Pnn)
:aa ,
(3.3) (3.4) (3.5)
and •
tt
•
(3.6)
t
dPnm/dt = -lCOnmPnrn + ( l / h ) Hnm (Pnn' -- Pmm)
where •
tl
.
t
.
IOOmn = lCOmn + Pmn :mn ; t
.
Pnn:nn =Fnn:nn --l-'nn:aa,
.
t
.
=From:ram
-
t
I'nn:mm = F n n : m m - F n n : a a ,
t
From:ram
tt
lOOnm=169nm + Pnm :nm ,
Fmm:aa
,
t
Fmm:nn
= Fmm:nn
-
Fmm:oa
•
(3.7) (3.S) (3.9)
366
R. Haberkorn et al. / Theory o f time-resolved level anticrossing experiment
In the following, we shall present our study of eqs. (3.3) - (3.6) for the time dependence of the system and the steady state behavior of the system. We first consider the solution of eqs. (3.3) - (3.6). For this purpose we shall employ the Laplace transform method. Applying the Laplace transformation to eq. (3.3) yields PPrnrn -- Prom(O) = -- ( I" / h ) . ( H m .n P n m. - .P r n n .H n m ). - I'm . : m. P m m - F m
:nPnn - P - l [ ' m :a ,
(3.10)
where Prom represents the Laplace transform of prom(t), Prom = P r o m ( p ) =
.;
e -vt
prom(t)dt
(3.11)
o
and F;n
-- r r 'n m
: Fn
t
; rn
ol
t
.
rm :n = F m m :nn ,
,
P m :a = I~rnrn :aa •
(3.12)
.
(3.13)
Eq. (3.10) can be rewritten as t
t
--
•
t
t
(p + rm :m) V m z + r m :nOn. -- (iHi, m/h) tim. + (iHmn/h) -~.m = Prom(O) - p - I r m :a . Similarly we obtain t
- -
t
Cn :rn P r o m + (P + r n
--
:n) P nn
•
t
--
.
+ ( i H n r n f f t ) Pmn •
ft
-
!
( i H m n f f t) P n m = P n n ( O ) - P - l [ ' n
- -
:a
(3.14) (3.15)
- (W'mnlh) ~,.m + (g4Lnlh) ~n. + @ + ~ , . n ) P.,n = p . , . ( o ) and .
tt
--
(3.16)
(iliUm/h) Prom - (iH~ml h) Pnn + (P + lWnrn) Prim = Pnm(O) •
From eqs. ( 3 . 1 3 ) - (3.16), we can solve forpmm, Onn, Pmn and~nm. For example prom(O) 1 ~?1"1
=
~.(p)
-
-
rm :JP
Fm :n
-iH'm/h
iHmn/h
Pnn(O) - I'n :alp
P + I~n :n
iHPnm/h
-iHmn/h
;ran(O)
iHm.l~
p + iW~nn
0
Onto (0)
-itInm/h
0
p + iWnm
(3.17)
where t
P + r m :m t
A4@) =
t
.
I-'rn:n
t
--iHnmff l t
.
t
.
t
iOmnffl .
t
Fn :m
P + Fn:n
iHnm/h
-itt'mn/h
-iHmn/h
itt'mn/h
p + iCOmn
0
iHnm/h
-iHtnm/h
0
P + iCOnm
(3.18)
It will be shown that this will lead to the equations which are commonly used in analyzing the LAC experimental data. 3.1. Case I - build-up
Here we study the build-up and steady state behavior of the system and in this case, we make the following approximations. The system is initially in the ground state a and the pumping is described by a rate constant, where state n is pumped by 1-'n:a and state rn is pumped by Fm :a- Further we assumed that there is no transition
367
R. Haberkorn et al. / Theory of time-resolved level anticrossing experiment
between n and rn other than that caused by the LAC. The decay rate of state n and m is equal to P; Pnn(O) = Pmrn(O) = Pmn(O) = ,Onto(O) = 0 ;
["n :m = rrn :n = 0 ;
Pn :n = Prn :m = [ ' .
Using these approximations, eq. (3.17) becomes Pm :a
(-1)
Prom
pn4fp)
--Prn :a
--i[tnm/pl
iHmn/h
Pn :a
P + r -- rn: a
iHnm/Pl
-'iHmnff1
0
iH'~dh
p + icO~nn
0
0
--iI-I;mffl
0
P + iWnmt
(3.19)
where P + l~ -- Pm :a
adp) =
-r. •
--I'm :a
:a
--il-l;m
p + p - r n :a
t
,
iH;,m/h
t
--iltrnn/h
iItmn/h
iHnm/h
-iHnm/h
il-Imn/h
.
.
--igm,,l~
tr
p + 1COmn
0
0
p + i~nm
(3.20)
The steady state solution of Prom (t) is given by r m :a
(-1) Prom (oo) - a4 (0)
- I ' m :a
-iI-I'nm/h
iHmn/h -iHmn/li
r n :a
F - Pn :a
iHnm/Pl
o
i~m,,Ih
ioo;;,.
0
0
--iH'nmffl
0
" "m 109n
(3.21)
Eq. (3.21) can be simplified as
Prom(°°)-
rm :a r'
(2 IHm. 1 2 / r ' h 2 ) ( r . , . / r ) ( r . :a - Fm :a) (COm.) 2 + r £ . + 4 r . , . I H m n l 2 / P h 2
'
(3.22)
where P '= F - Fn:a - Pm:a •
(3.23)
Eq. (3.22) describes the steady state population and the lineshape for coherent excitation of levels rn and n (coherent excitation here means Pnm(°°) --/=0; this is the case whenever I'n :a :~ Pm :a) and is commonly used in analyzing the experimental data of the steady state LAC experiment [2]. But as can be seen from the approximations stated above, eq. (3.22) is valid only when P n :m = F m :n = 0 and l-'n :n =Fm :m = PThe same result can be obtained by setting dPmm/ dt = dpnn/ dt =' d p m n / dt = dpnm / dt = 0 .
(3.24)
Next we shall study the time dependent behavior of Prom (t). It can be expected that in this case one would obtain the time evolution of the build-up of Pmn(t), Pmm (t) and Pnn(t). Notice that eq. (3.19) can be simplified as -ffmm
_ (--1)[Fro :a(P + P)(P + i6°mn)(P + iC°nm) + (21H'~nl2/h2)(P + I'mn)(I'm :a + rn :a)] p(p + r - I~m :a - Pn :a)[(P + F)(p + iWmn)(P + i ~ n m ) + (4 [Hmn 12/h2)(P -I" Pmn)]
(3.25)
or ( - r m :a) [(P + r ) ((p + rmn): + ( ~ m . ) : ) + (2 la~,. 1 2 / h : ~ + rmn)(1 + r . :a/rm :a)] ffmm - p(p + r - r m :a - rn :a)[(P + r ) { ( p + rmn) 2 + (ag~n} + (4lH~nnl2/h2)(P + rmn)]
(3.26)
368
R. Haberkorn et al. / Theory of time-resolved level anticrossing experiment
Here the following relation has been used •
rt
.
tt
t
(19 + ICOmn)( p + 16grim) = (]9 + r m n ) 2 + (6)mn)
2
(3.27)
.
If 1" = Pmn, then eq. (3.26) reduces to Finn) 2 + (LOmn)2 + (21H'mnl2/h2)( 1 + Ian:a/Fro : a ) ] p(p + I" -- I~m :a -- I~n :a)[(P + I~mn) 2 + (O)mn) 2 + 4Ill'ran 12/h2]
(--Fm
-Prom
:a)[(P
+
or
r
Prom -
(--I'm :a)[(P + Pmn) 2 + (6°mn)
2
+ (2]Hmnl2/h2)( 1 + P , : a / r m :a)l
p(p - x,)fp
- a2)(p
- x3)
(3.28)
,
(3.29)
where X3 = - ( P - P m :a - Pn :a) and )t~ and X: represents the two roots of the equation t
2
t
(p + Finn) 2 + (¢.Omn) +4[Hmnl2/h 2 = 0 .
(3.30)
Carrying out the inverse Laplace transformation of eq. (3.29), we obtain Pmm(t) (--Pm
:a)
-
' 2 (_Pn:a _ 1 ) [ Pmm(°°) +2[Hmnl ~-m
:~-
h2
\ I ~ m :a
exit
e•2t
)klCA' - - ) k 2 ) 0 k l - - X 3 ) ~ )k2CA2 -
1
) k l ) 0 k 2 - - )k3)
eXat[()~ 3 + Finn)2 + (COm.) , 2 + (21H~nlE/h2)(1 + rn:a/rm :,)l +
)k3 (~k3 _ )k 1 )()k 3 _ ~k2 )
,
( 3 . 3 1)
where Pmm (oo) represents the steady state population of Prom(t), and is given by eq. (3.22). In other words, eq. (3.31) describes how the LAC signal reaches the steady state. The terms involving e M t and e M t give rise to an oscillatory behaviour of the build-up of the system.
3.2. Case H - decay Here we shall study the decay behavior of the system. For this purpose, we shall assume an arbitrary population and phase of the levels n and m at the beginning, that is,
Pmn(O):/:O,Pnm(O)#:O;
Pnn(O)eO, Pmm(O):/:O;
Pm:a=0, rn:a=O.
Experimentally this means that we have a coherent population at the beginning with the excitation source removed. In this case we have --
t
2
P
P nn = A4 1 (P)~0mm(0)[(P + I'n:n) [(P + ['mn) 2 + @Omn) ] + (21Hmnl2/h2)(P + Pmn)]
+p,,(O)[(21Hm, 12/h2)(P + rmn) - Pm:n[O + Fro,) 2 + (COm,)2 ] ] + Pmn(O)(iHnm/h)(P + r m :n + r n :n)(P + i6°nm + r m n ) + Pnm(O)(-iHmn/h)
X (p + rm :n + r . :n)(P + iCO~n + Finn)) ,
(3.32)
where r
2
A,O,) = [0, + r m . ) = + ((-Omn) ] [(l19 + I~m :m)(P + rn:n) -- rm :nrn:m]
+ (2[HmnlZ/h2)(P + Pmn)(2P - Pa:m -- Pa:n) •
(3.33)
Next we consider some particular cases of eq. (3.32). For example, if Pn :m and Pm :n are negligible and Prnn = Pn :n = Pm:m (no pure dephasing), which means that the transition between the states n and m is only due to LAC and that both states have equal decay rates, then eq. (3.32) reduces to Pmm(P) =-pmm(p)l + Pmm(P)2 ,
(3.33a)
R. Haberkorn et aL / Theory of rime-resolved level antierossing experiment
369
where Pmm(P)l =
Pmm(O)[(P + Pmn) 2 + (C~mn) 2 + (21Hmn 12/h2)] + (2triton
I=/h2)p,m(0)
(3.34)
(P + Pmn)[(P + Pmn) 2 + (~°m,) 2 + 41H'mnl2/h21
and (llfl)[Pmn(O) Hnm( p + i~3'nm + I'mn )
' Pnm(O)Hmn(P +"lCOrnn , + rmn)] ,2 , (1° + r m n ) [ ( P + l~mn) 2 + (6Omn) + 41H'mnl2/h2]
•
Prom(P)2 -
,
_
(3.35)
Notice that-Prom (P)l and Prom (P)2 are due to incoherent and coherent initial population, respectively. It follows that (3.36)
Pmrn(t) = Pmrn(t)l + prom(t)2 ,
whereA2m,= (corn,,) ' 2 + 41H'mnl2/h 2 , r 41Hmn I2 e - r m n t sin 2 ~Amn t (bAron) 2
e-I'mn t Pmm(t)x = Prom(O) -(hArnn)2 --[(hCOmn) 2 + 4 1 g m' , I 2 cos2½Amn t]
+Pnn(O)
(3.37)
and 4COmne-Pmn t Prom (02 . . . . . . . . . . . . Re [Omn(O) Hnm I sinZ ~ Amnt hA2m,
2 e - rmn t
t
Im [ p m , ( 0 ) H , , ,
hAm,
]
•
(3.38)
Pnn(t) can be obtained from the expression of Prom (t) by interchanging indexes n and m. The LAC system will exhibit oscillatory decay with the beating frequency given by Amn. Notice that if both levels have equal initial populations [Pmm (0) = Onn(O)J, then eq. (3.39) becomes
Prom (01 = Prom (0) e x p ( - P m n t ) .
(3.39)
That is, in this case, the oscillation in Prnm(t)l disappears. But if Pmn(O) ¢ 0, which relates to an initial phase relation between the two levels due to a coherent excitation, the beat can still take place through Prom (t)2. In particular, if ann(O) = 0, and Pmn(O) = 0, then eq. (3.36) reduces to Pmm(t)/Pmm(O ) = ( e - r m n t / A ~ n ) [(Cdmn , ) 2 + (41Hmnl2/h , 2) cos2~Amnt] . t
(3.40)
2
If(cOmn) < < 41H'mn IZ/h 2, then eq. (3.40) reduces to Pmn (t)/Pmm (0) = exp(--Fmn t) COS21~Amn t ,
(3.41 )
where Amn ~ 21Hmnl/h. This represents the case for the LAC experiment near the crossing point. The decay will be nearly 100% modulated and the beat frequency will be the direct measurement of the interaction strength. On the other hand, if t2 a~mn >> 41H'mn 12/h2, then (3.42)
Pmm (t)/Pmm (0) = exp(--I" m n t ) .
That is, the experiment is far from the crossing point and the oscillation term is negligible in this case. 3.3. Numerical calculation For convenience, we introduce the dimensionless quantities *
*
t = (Fn:n/I'n:n) t ,
¢
,
i
*
(Hmn) =Hmn(Fn:nlI'n:n) ,
t
,
r
,
(Oanm) = ~ O n m ( r , , : , / r , : , )
etc .
In figs. 1 and 2 we show the results of the time-dependent behavior of the build-up and the decay in the LAC
R. Haberkorn et al. / Theory o f t i m e - r e s o l v e d
370
• 5 -i
antierossing experiment
:MRG [RHO]MN
z G~,
.5
IMBG
.5
RERL [RHO)MN
[BHO]MN
W ----2--222:::: . . . .
C-)
CO
level
co:: : .
. . . . . . . . . .
- -
:33 ~2 CD
.5
RERL
[RHO}MN
FTI rCo
~ ~.~ 212 2E cD
ED 7~ Z .5
--7
[RHO}NN
2I) ZI2 ~
CD Z 7
-.5.
(RHO]NN
1.0
.5
,1
(RHO]MM
~
.3 ,
1.0_
[RHO]MM
_
22g
.1
0.0
.... ..:L I
~.~
,5
........
I 1.~
I 1.5
I 2.0
I 2.5
0.0 0.~
3.1~
.5
(r,,: a, COnm
= =
1): 5.0.
evolution of LAC experiment, build-up case .... Oan*rn = 0.5, - - Wnrn = 1 . 0 , - -
1.5
2.0
2.5
3.0
TIME
TIME Fig. 1. Time
1.0
F i g . 2. T i m e e v o l u t i o n
(Onn(O)
=
of LAC experiment,
decay case
1): .... Wn*m= 0.5, - - - - oan*rn= 1 . 0 , -
eOn*m = 5.0.
experiments, respectively, for the case Pn:n = Fm :n = 1 , (COnrn) '(H'mn) I, and Fm :n = Fn :m = 0. Notice that for the build-up case we have assumed that P~, :a = 0 and Fn :a 4: 0, i.e. only the n state is pumped. In other words, it is assumed that the transition a -+ m is forbidden. The purpose of figs. 1 and 2 is to show the effect of the decay rate constant on the time-dependent behaviors of the LAC system. Numerical calculation for many other cases has also been performed, but the qualitative behaviors are quite similar to those shown in figs. 1 and 2. In figs. 1 and 2, we show not only the time evolution of the populations Prom (t) and Pnn(t) but also the time evolution of the Re [Pmn(t)] and lm [Pmn(t)], the phases of the system. As can be seen from these figures, the quantum beat becomes pronounced when Icomnl/Fn: n > [(or IHmnl/i'n:n > 1). When we want to classify the LAC experiment we have to consider three cases, which are related to the magnitude of the damping Fn :n of-the system compared to the interaction leading to LAC. In figs. 3 a - 3 c , we show the time-dependent behavior of Pnn and Prnrn for the three cases, IHmn[ > I'n :n' IHmnl = Pn :n and IHmnl < Pn :n" Here for simplicity we have assumed f m :m = 0, I"m :n = Pn :m = 0. This means the system decays only via Pnn and there is no pure dephasing of the system. The calculation is performed for the decay of the system, and only state n is populated at the beginning. The time-dependent behaviors of Pnn and Omrn for Pn :m :/= 0, I'm :n =/=0, and I'm :m 4:0 are qualitatively similar to those shown in figs. 3 a - 3 c . Besides the excitation process the experimentalist can adjust the external field strength which will result in the energy gap change. Thus in figs. 3 a - 3 c we also show the effect of changing the energy gap (i.e., the scanning of the magnetic field in the conven=
R. Haberkorn et al. / Theory o f time-resolved level anticrossing experiment
,.o_
{RHO]NN
"'\X " " " " ,
Z
"-'\
(RHO]iNN
1.0_ //\
•
371
..
o[/X/X/;'4.'~ ] 1
0.0
.5
I
''
l.~
[
' ~'
1
L
', 1 "\
2.El
,.5
2.5
" 3.
TINE
C
ioL
[RHO)NN
5
[RHODMM
1.0_
[RNO]MM ZZI3
(D
.5
/ \, O.fl
0.0
.5
0.0
,.0
,.5
2.0
~.~
~.o
.5
O.t~
1.(3
TIME Fig. 3. T i m e e v o l u t i o n o f L A C e x p e r i m e n t , 5.0,
rnn
=
1.5
2.~
2.5
3.2
TIME = 5 . 0 ) : ( a ) Fmm * = 0 . 0 , Fnn * =1.0;.. . . Wnm '* =0.0,--tOnm '* = ' * --- 0.0, -- -- -- tOnm '* = 5.0, ' * - 2 5 . 0 . ( c ) Fmm tOnm tOnm * = 0.0, ~nm = 25.0.
d e c a y c a s e ( H n ,' *
* - 5.0;.... t o 'n* , - 2 5 . 0 . ( b ) r m* m = 0 . 0 , I'nn 1 0 . 0 , . . . . t°nm = 0 . 0 , - - - - t ° n . = 5 . 0 ,
tional LAC experiment) on the density matrix elements. As can be seen from these figures, as the energy gap increases, the quantum beat becomes less pronounced. 3.4. Further discussion o f the steady state case
Notice that in obtaining the steady state solution given by eq. (3.22) two assumptions are made, i.e. rm :n = = 0 and r m : m = rn:n = r. Without these assumptions the steady state solutions of eqs. (3.3) - (3.6) are given by rn:m
t
r-:a(Zmn P,m
(Zmn - rm:m)(amnr. :a(Amn
Pnn
- r n : n ) + r n :a(Amn + r n : n )
-----
r
rn:n)-
(Am. _ rm :m)(Am. -
(Ann + rm:n)(Am.
r ; . :m) + r .
: a ( A m n
? t r.:n) - (Amn+ r m
r~ :m) :n)(Amn +
,
,
(3.43)
+
(3.44)
372
R. Haberkorn et al. / Theory o f time-resolved level antierossing experiment
and t
.
t
Prnn
= (i/h) H r n n ( P m r n -- Pnn)/(l~Ornn + P m n ) ,
(3.45)
Amn
= -2I'mnlH~12/[(hcO'~n)
(3.46)
where 2 + ( h F m n ) 2] •
In other words, eqs. (3.43) - (3.45) give us the exact results of the steady state population and the phases of the three-level LAC system. Other particular cases can be deduced from these equations. t t f f It should be noted that in some cases Hna (or Han ) a n d H m a (or Ham) may not be zero, but the effect of these terms is negligible as long as the energy gaps COna and COma are much larger than I H a m 1, I H a n l , and I H n m l (see appendix).
4. Fivedevel model 4.1. Case I
Next we consider the following five-level system which represents the case for the anticrossing of two doublets connected with the same ground state: (D~)
.n'
(D1)
n
(D2)
m
(D~)
m'
(So)
a
n and m are the levels in which the anti-level crossing takes place. In other words, we have only/~/~nn ~ 0 and the other matrix elements o f / ~ ' are zero. In this case, from eqs. (2.2) and (2.3), we find dpmrn/dt
= -
" ) ( n m.n P n m (l/h
-- . P m n H n m ). - P m
. ' m ' -- P m :n Pnn -- P m : n ' P n ' n ' -- P m :a , :.m P m m -- . F m : m ' P m
(4.1) dPnn/dt
= - ( l"/ h ) ( H n m ' Prnn - P n m H m' n )
-
P n. ; n P n n .- r n :n'. P n ' n ' --. P n :m Omm -- F n : m ' P m ' m
' -- Pn :a ,
(4.2)
= - (it°ran +
dprnn/dt dOn'n'/dt dpm'm'/dt
-- (i/h)Hmn(Pnn - P m m ) ,
Pmn)Pmn
r
(4.3)
t
= -Pn'=n'Pn'n'
- rn':nPnn
= - - I ~ m' ' = m ' P m ' m
- P~':m Prom
' -- ~ m' ' : m P m m
-
-
rn':m'Pm'm'
-
rn':a ,
-- F m' ' : n P n n -- F m ' : n ' ton ' n ' -- P m ' : a
(4.4) (4.5)
by eliminating Paa and an expression similar to eq. (4.3) can be written down for O h m . Let us first consider the steady state case. The steady state solution for the pseudo four-level system is given by 1
Prom(°°) =~
I ['m:a' Pn :a
-(Amn+Pm:n+P~n:m'Rm'n+P~n:n'Rn'n) Am.
- r; :. - r;:~'Rm'n
I
(4.6)
- r;:n'Rn'n
and !
1
Pnn(°°)
=~)-2
Amn
-I'm:m
-(Amn
-Pm:m'Rm'm
+ Pn :m + r;, : m ' R m ' m
-Pm:n'Rn'm
+
r;,:.'R.'m)
Pm:a
r . :a '
(4.7)
R. Haberkorn et al. / Theory o f time-resolved level antierossing experiment
373
where t
t
¢
-(Amn
-(Amn + rm:n + r~, :..'Rm'n + rm :n'Rn'n) I
t
Arnn - F m : r n - F m : m ' R m ' m
D2 =
- Fm:n'Rn'm
F
t
A mn - ['n:n -- r n :m ' R m 'n - r n : n ' R n ' n
+ Fn: m + F n : r n ' R m ' m + F n : n ' R n ' m )
t '
(4.8) --1
Rm, m = A 2
r
t
t
(Fm':n'I'n':m
t
-- F n , : n , P m , : m ) ,
(4.9)
n :n I r a ' : n ) ,
(4.10)
Rm'n = A~l(pm':n'l~n':n
(4.11)
R n , m = A ~ - l f p~.--n ' , :m ,r, r , :m - - ~ m' ' : m ' F n ' ,: m ) , --in t
t
t
t
R n ' n = A { 1 ( F n ' : m '['rn':n -- F m ':m ' I ' n ' : n )
(4.12)
A.,n = - 2 r m n IHmn 12lh [(~mn) ~ + r ~ n ] ,
(4.13)
and
t
!
t
with A= = P~n' :m'l~n ' :n' -- Pro' :n'Fn ' :m'. It should be noted that in the literature I~,n :m' ~ F m : m ' , I"rn :m ":" Pm :m, Fn :n ~ l?n :n' etc. have been assumed. That is, the contributions from the pumping rates r m :a, r n :a etc. have been ignored [cf. eqs. (3.8) and (3.9)]. Derouard et al. [10] have also studied this system, but the following approximations have been made by them: (a) no cross-relaxation between different doublets t
!
~.
t
. . m =. r n . . m., = F n. , = m = . Fn,:m, = Fm,:n . FIn:m =I~n:rn ' =. F m : . n = F .r o : n = F r o : n' = I'n.
t
= F m ' : n ' = 0 " (4.14)
(b) equal lifetime of the states, i.e. l~m :m = I~m :m = I"n :n = rn :n = F n' , : n , = F m, , : m , = - F o - F l ;
(4.15)
(c) equal relaxation rates within the doublets t
rn:n'
t
= rn :n' = rm :m' = Fm'=m
= rl
•
(4.16)
Using these approximations, eq. (4.6) reduces to
rm :~ Prom -
~t
(rm +
:a --
rn :a)
21a~nnl 2 ,
"Y
(h¢Omn)
2
+ (hPmn) , + (4Pmn/m)lHmnl2
,
(4.17)
where 7 = - P o (Yo + 2P1 )/(Co + P x), which is equivalent to the expression derived by Derouard et al. if we make the identifications, Po = _~/(o) and P1 = - ~ ( ' r (1) - 7(0)). 4.2. Case 11
Another five-level model to be considered is
(Sl)
n
(T+ 1)
m
(To)
m'
(T_I)
m"
(So)
a
This corresponds to the anti-crossing between the singlet state and the triplet state. In this case, the master equa-
374
R. tfaberkorn et al. / Theory o f time.resolved level antierossing experiment
tions for the pseudo fourqevel system are given by d p n n / d t = -(1h/)(HnmPmn"
'
- PnmHmn)'
- P n. : n P n n -. P n : m .Prom - .P n : m ' P m ' m '
- Pn:m"Pm"m"
- Pn:a ,
(4.18) dPm m / d t = - ( i / h ) ( H m n P n m
- P m n H n-m )'
rm' : m P m m
-
.Pnn
rm:'
rm'
-
:m
'Pro ' m '
•
t -- Fm :m "Pro "m"
-
dPmn/dt = -(i~mn
(4.19)
P m :a ,
+ P m n ) P m n - (i/h) H m n ( P n n - P r o m ) ,
?
!
(4.20)
t
p
4.21)
d P m ' m ' / d l = - - I ~ m ' : m ' P m ' m ' -- I ~ m ' : m " P m " m '' -- F m , : m P m m -- F m ' : n P n n -- F m ' : a
and .
n
~- _
dPm m / d t
?
.
Fm
.
.
.
:m P m m
t
t
?
- I~m":m'Pm'm ' - r m " : m P m m
(4.22)
- ['m":nPnn - I'm":a •
Let us first consider the steady state case. F o r the case Fro' :a = 0, Fm" :a = 0, we find 7 Fm "a •
? : n + F mp : m ' R m ?' n + ['m
-(Amn
Prom = ~ - ,
1J2 [ I'n :a
Amn - rn
:n
+ F mt : m " R mr" n )
, , - Pn : m ' R ' m ' n - r n : m " R m " n
(4.23)
and ?
= 1 Pnn
t
Amn-Fm:
-~2
F
t
r
m -Pm:m,Rm,m-['m:m,,Rm,,
- ( A m n + I ' n : m. + V n.: m ' R.m ' m +. I ' n : m
m
F m :a rn:a
"R~n"m )
(4.24)
,
where
_(Amn - r m : ~
D' 2
--
- P~n:m'Rm'm
F
]-(Amn
-1-'~:m"R~",n
t
F
+ Fn:m + Pn:m'Rm'm
-(Amn
t
l
+ ?m:n t
+ rm :m'Rm'n t i
Amn - Pn'n - Fn:m'Rm'n
+ Fn:m"Rm"m)
+ rm:m"Rm t ¢
"n)
- I"n:m"Rm"n
(4.25) F
Rm'm
= (1/A~
t
.
)(rm
':m"
.
.
t
t
~
(4.26)
I"m,:m ) r
Rm'n = (1/A2)(rm ':m"r~" I
- -.I " m , , : m , ,
t
¢
:.
- Vm" :m " F i n ' : n ) ,
t
?
(4.27)
t
(4.28)
R m , , m = ( 1 / A 2 ) ( P m , , .. r e , I ' m , . , m -- F m , : m , F m , , : m ) t
t
I
t
f
t
:n - F m ' : m ' F m " :n)
Rm"n = (1/A2)(Fm" :~'r,,'
(4.29)
and Pm':m'
Pm":m
"-
Pm" .mIFrnr.m ,
(4.3o)
.
Notice that r
Pm'm' = Rm'mPmm
(4.31)
+ Rm'nPnn
and f
r
Pm"m" = Rm"mPmm
.
(4.32)
+ R m nPnn •
In particular, if one uses the a p p r o x i m a t i o n s [see eqs. (4.14) - (4.16)] t
t
?
r~:m = r ; . : n = r L ' : n = r n : m ' = r n : m " = r m " : n -- 0 , , Fm :m' = I'm':m
= F m p :m
,,
= F r ro " :m = F mr ' : m
"
=
r m "
• m'
(4.33) =
r
1
(4.34)
R. Haberkorn et aL/ Theory of time-resolved level anticrossing experiment
375
and P;n: m = ['m':m' ' ' ' --=- P o - Pl , = 1"m":rn" = Pn:n
(4.35)
then the steady state solution can be simplified as Amn(1"m :a + 1"n :a) + 1"m :a(Co + 1"1) '°ram = 2Amn(ro + I"1 - 1"]/Po) + (Po + r l ) ( F o + Pl - 2F]/Po) Pnn -
Amn A m + Fo + F1
Pmrn +
Pn:a A m n + 1"o + 1"1
'
(4.36) (4.37)
,
Pm 'm' = Pm"m" = (1"1/1"o) Pmm •
(4.38)
This gives us the linewidth and the steady state population of the system. The time-dependent behaviors of the above two pseudo four-level systems can be studied by solving respectively eqs. (4.1)-(4.5) and (4.18)-(4.22) by the Laplace transform method. We have also performed the numerical calculation to study the time dependent behaviors of these two models. Since the results are qualitatively similar to those of the pseudo two-level system, they will not be presented here. Although the steady state and time-resolved LAC experiments can be qualitatively understood by using the three-level model, for the purpose of quantitatively analyzing the LAC experimental results to determine the LAC experimental results to determine the LAC coupling constant, decay rate constants and dephasing constants, it is necessary to treat the physically more realistic models like those presented in this section.
5. Discussion
In the previous sections we have presented a theoretical approach for the systematic treatment of steady state and time-resolved LAC experiments. It should be noted that in deriving eq. (2.1) which is our starting equation, the Markov approximation has been made, the singular perturbation method has been used to second order and the system and the heat bath are assumed to be decoupled initially. In the time range where the Markov approximation is not valid (i.e., the transient region) oscillatory decay may be observed depending on the nature of the memory function involved in the relaxation processes. However, the quantum beat discussed in this paper is mainly due to the interference between the LAC states and not due to the non-markovian effect. Conventional LAC experiments are carried out under steady state conditions; in this paper, we have treated the time-resolved LAC experiment. In this connection, it is often convenient to work in the frequency domain rather than in the time domain. In that case, we need only to carry out the Fourier transformation of the results for prom(t), Pmn(t) etc. obtained in the previous sections. For example, the Fourier transform of Prom(t) given by eq. (3.40) can be expressed as
Prom((0) prom(O)
-
2P,..
'
(0mn)
Am n2
2 +21H,,ml 7 1 IHmnl r + ~ J (02 + F2n ~ L ( ( 0
1
+ mmn)
2 +
1"ran
+
1
((0 - mmn) 2 + r2mn
(5.1) which consists of the superposition of lorentzians. We have shown that the quantum beat can be observed in the time-resolved LAC experiments under suitable conditions. From the pseudo two-level model, we can see that favor able conditions for observing quantum beats are [see for example, eq. (3.40)] 2
(5.2)
Amn >~ PZmn
(the dephasing rate constant is smaller or comparable to the beat frequency) and r
41Hmnl
2
i
~ h((0mn)
2
(5.3)
R. Haberkorn et al. / Theory of time-resolved level anticrossing experiment
376
c'*= 2H"
II
t
2H* Fig. 4. Quantum beat conditions.
(the LAC interaction is comparable or larger than the energy spacing of the unperturbed levels). These conditions are shown in fig. 4. In this figure, we have plotted ¢Omn/I~rnn against 21H'mnI/hFmn. The favorable conditions for observing quantum beats are shown in region I surrounded by the heavy lines. Notice that in making the plot of fig. 4, it has been assumed that Pn :n = I'm :m" To see the effect of Pn :n 4: Pm :m on the observation of the quantum beat, we consider the case of COmn = 0. In this case, eq. (3.17) for the decay of the system after removing the excitation source can be written as ,
Prom = ~°mm(0)[2h -2 Inmn I + (p + rn :n)(P + rm :m)] t
2
t
+ 2h -2 Inmnl Pnn(O) + 2(e + Fn :n) Im[HmnPnm(O)] } , 12 - ~(Fm :m - Pn:n)Z]}-i × {(P + Fmn)[(P + Finn) z + 4h -2 Inmn
(5.4)
Here it has been assumed that Pm :n = Pn "m = 0 and r(d) = 0; i.e., the pure dephasing is also zero. Carrying out • Iron the inverse Laplace transformation of eq. (5.4) yields
Pmm(t) = Pmm(O) e-rmnt~L2h2fm nH l ~nnzl+ (1
~]coslHmnlZ]2~mn t + (Fn2~mn:n--
Finn)
sin
2~mn t]
+ Inmnl~ Pnn(O) e-trent(1 - cos 2~mnt) + 2 Im[(Hmn/h) Pnm(O)] e -rmnt 2h 2fmn 2~mn sin 2~mn t ,
(5.5)
where ~mn 2 = h - a IHmn , I2 - ~ ( F n :~ - Finn) 2 • Eqs. (5.4) and (5.5) indicate that the quantum beat disappears if /3:mn < 0 or (rn:n -- rmn)2/16 > IH'mnl2/h2. The theoretical approach for LAC presented in this paper takes into account not only the decay rates due to radiative process, radiationless transitions and collisional processes, but also the dephasing rates due to inelastic and elastic (or pure dephasing) processes. It should be noted that in calculating 1a to the second order approximation with respect to HRrq, the coupling between the system and the heat bath, the collisional rate constant obtained from expressing the matrix elements of P in terms of the concentration of the collision partner corresponds to that obtained by using the collision theory in the first Born approximation [11,12]. Recently, quantum beats have been reported by Chaiken et al. [13] for biacetyl in the gas phase. We may qualitatively interpret their results by using the pseudo two level model [see for example eq. (3.40) and figs. 3 a - 3 c ] . l* r t . . In this case, we have IHmnl > Pn :n'mmn >~>Finn and h6onm < IHmnl; I.e., the luminescence decay is much slower than the oscillation (14). In the theoretical treatment of the steady state and time-resolved LAC experiments presented in the previous
R. Haberkorn et al. / Theory o f time-resolved level antierossing experiment
~.a]
377
POPULRTION OF STRTE N
233
5.5
L/_.-
@.
- .......... I
1.@
]
I
2.0
3.@
- . . . . . . . . . . . . . . . . . . . . . . . . . . . . I
I
~.~
5.@
I
6,1~
I
7.1?
I
8.8
9.
TIME
Fig. 5. Effect of incident light on LAC experiment. The light intensity is expressed in terms of the Rabi frequency I~E;From = O, rnn = 1, Hmn = 1, COmn - 1; ... is.E= 0.6, - - - ~LE= 2.0, ~E = 6.0.
sections, the pumping o f the system has been described by the rate constants like Pnn :aa, r m m :a~, etc. Suppose that the pumping in the LAC experiment is carried out by radiation. In this case, in addition to H ' involved in eqs. (2.1)-(2.3), we need the interaction between the molecule and radiation which in the dipole approximation can be expressed as (5.6)
[1; = - It" E ( t) ,
where It represents the dipole operator and E ( t ) is the time-dependent electric field vector of the incident radiation. Here we have used the semiclassical theory for/lr- Using eq. (5.6) one can study the effect of light intensity, excitation duration, excitation bandwidth, Doppler broadening etc. on the LAC experiment. For example, in fig. 5 we show the effect of light intensity (which is expressed in terms of the Rabi frequency/.rE) on the build-up of the populations in the LAC experiment for the case Itma 0 and Itna ~s 0 (i.e., only the n-state is dipole-allowed). As can be seen from fig. 5, the time-dependent behaviors of the system are sensitive to the incident light intensity. It should be noted that if the time scale is in microseconds, the unit of the Rabi frequency is MHz and the value of 6 corresponds to dye laser pumped by a cw argon ion laser [15]. It is important to note that in the steady state LAC measurement one can only obtain the position and width of the spectra [see, for example, eq. (3.22)], while in the time-resolved LAC measurement one can obtain information like beat frequency, relative decay (and/or build up) amplitude and decay constant (see for example figs. 1 - 3 ) by changing the energy gap (sc~inning the magnetic field strength). In other words, whenever possible, it is useful to t carry out both steady state and time-resolved LAC experiments. In this way, one can determine not only H m n but also rate constants I~mm :ram, From :nn etc. and dephasing constants Finn :ran etc. =
Acknowledgement Financial support of the Alexander-von-Humboldt-Foundation, the NATO Scientific Affairs Division and the Deutsche Forschungsgemeinschaft is gratefully acknowledged.
378
R. Haberkorn et al. / Theory o f time.resolved level anticrossing experiment
Appendix t
In this appendix, we shall study the effect of nonvanishing H~n, Hna on the steady state solution of the pseudo two-level system. For this purpose, the perturbation method will be employed, •
t
(1/h)[X(HnaPa n
t
-
-
!
•
F
r
t
t
!
r
+ r . : . p . . + Fn:mPmm + I'n ."a = 0 ,
PnaHan) + (HnmPm n -- P n m H r n n ) ] t
t
(1/h)[X(HmaPam - PmaH'am) + (Hmn Pnm - PmnHnm )] + r m :mPmm + r m .., P.. + rm .-a = 0 , t
(iCOmn + Finn) amn + ( i / h ) [X(HmaPa n - PmaHan ) + Hmn(ann - P m m ) ] = 0 , (~lOma" '
+ V m a ) P m a + (i/lT)[XHrna(Paa'
(iCO~a + F . a ) P n .
t
(A.4)
t
+ (i/h)[Xl-Ioa(P~a -- P n . ) + n n m P m a
-- )kPnrnnma]
(A.2) (A.3)
- XPmnH~a] = 0 ,
- P r o m ) + HmnPna'
t
(A.1)
t
(m.5)
= 0,
where X represents the perturbation parameter. Pnn, P m m , P m n , Pan and Pare have to be expanded in power series of X; for example for Pnn, we have = t~ r l (F/yl 0 ) "1" )'t~(1) ' "t" n r / + ~k2
Pn/2
p(2n) +
(A.6)
....
In the zeroth order approximation, we find p(O) _ _ (o) _ 0
net - Pma -
(A.7)
,
and t.,nn^(°),p(Om) and Pmn(°) are given in section 3. In the first order approximation, we find p(1) - n(l) (1) nn - ~'rnm = Pmn = 0 ,
(iCOma+ •
'
(A.8)
i, ma)p(lm) + (l/h) • [nmaCOaa ' (0) -- H.(0) m m . ) + n m n P n a(1) -- p(mO~/~a] = 0
(a.9)
'
and
(ieO~a + r.a)p~ )
+ (i/h)[H~a(p~) -
-
,,(o)): + /"4' n, m t ~n(1) V'nn m a _ en(o)/4, nm"ma] ] = 0
(A.10)
•
Eqs. (A.9) and (A.10) can easily be solved
n(1)=g;/h~rH' (n(o) r~na ~1 : t na~'aa
-
-
^(o)): - HmaPnm]/(lCOna ' (o) " ' + Pna) t~nn
(A.11)
and Pma ,.,(0) 4 ' n(O) 1/(i,.,' + P m a ) • (1) = (i/h )[_H~na(p(Oa~) _ ~" m r n ,~ + /"'na~" mn,tk'wma
(A.12)
Next we consider the second order approximation, • t (1) fit r~(2) (1/h)[(HmaPa m - p(lm)aHam) + [v'rnnt-'nm
n(2)/4t
~] " [ ' I ~ t
t'mn"nmzJ
(i/h )[(HmaP(al)rn - p(ml)aH'am) + (/4'v"mn~" nrn~(2)_ ~'rnn--nm'/~4(2)'
n(2)+r'
nlm-'nn
r~(2)
--n:mt~mm
=0
,
2) +pt ~(2) = 0 :J3l + I ' m, :m p (m m - m :nr'nn
(A.13) (A.14)
and • '
OCOm
n
(2). + (i/h)[fH~naP(aln)- p(')H'ma__an)+/4'--mn~s'nn(n(2)- p(m2L)] = 0 . + r m . )-P m
(A.15)
F r o m eqs. ( A . 1 1 ) and ( A . 1 5 ) , w e can see t h a t the c o r r e c t i o n terms P~)n, Prnm-(2)and Pmn(2)depend on Pna(1)and PmaO) ¢ t which in turn depend on the energy gaps COnaand ~)ma. In other words, if the energy gaps are large compared with the magnitude of H~n and H~n, the contribution to [ n n , Prom and P m n from H~n and H~m is small (i.e., the second t t order with respect to Han and H£~n). Explicitly t"nn~ A2) Pmrn ^(2) and P^(2) m n can be found from eqs. (A.13), (A.14) and (A.15),
p(2) mm = { ( B m n -- P m : n ) [ C m n + 2h -1 Im(P(n~t/an)] - (Bran + r n': n ) ( C m n x
[(Sm.- r ; ,
:n)(Bmn - P n : m ) - (Bmn + r n : n ) ( B m n +
rm : m ) ]
-1 ,
-- 2h -1 Im(P(ml)H'am)} (A.16)
R. Haberkorn et aL / Theory of time-resolved level anticrossing experiment
379
p
t
r
× [(B,,, - V, :m)(Bm, - I'm : , ) - (Bin, + I', :,)(Bin, + I'm :m)]-t
(A.17)
and O(m2)ncan be obtained from substituting eqs. (A.16) and (A.17) into eq. (A.15). Here Bran and Cmn are defined by Bmn = 2I~mn IHmn 12/[(ho3rnn) 2 "t" (h Fmrt) 2 ]
(A.18)
Cmn =2h-2 Re [Hnm , , ( 1 ) -- PmaHan)/(lO3mn (1) , • ' + I'mn)] (amaPan
(A.19)
and
References [ll T.G. Eck, L.L. Foldy and H. Wieder, Phys. Rev. Letters 10 (1963) 239. [2] H.J. Beyer and H. Kleinpoppen, in: Progress in atomic spectroscopy, eds. W. Hanle and H. Kleinpoppen (Plenum Press, New York, 1978), Part A, pp. 607-637. [3] V. Macho, J.P. Colpa and D. Stehlike, Chem. Phys. 44 (1979) 113. [4] L. Benthem, J.H. Lichtenbelt and D.A. Wiersma, Chem. Phys. 29 (1978) 367. [5] J.A. Mucha and P.W. Pratt, J. Chem. Phys. 66 (1977) 5356. [6] D.H. Levy, J. Chem. Phys. 56 (1972) 5493. [7] T.A. Miller, J. Chem. Phys. 58 (1973) 2358. [8] S.H. Lin and H. Eyring, Proc. Natl. Acad. Sci. US. 74 (1977) 3623. [9] R.I. Cukier and J.M. Deutch, J. Chem. Phys. 50 (1969) 36. [10] J. Derouard, R. Jost and M. Lombardi, J. Phys. Letters 37 (1976) L135. [11] R.D. Levine, Quantum mechanics of molecular rate processes (Oxford Press, London, 1969) pp. 70-78. [12] M.L. Goldberger and K.M. Watson, Collision theory (Wiley-Interscience, New York, 1964) pp. 344-383. [13] J. Chaiken, T. Benson, M. Gurnick and J.D. McDonald, Chem. Phys. Letters 61 (1979) 195. [14] Detailed analysis to be published. [151 K.E. Jones, A. Nichols, and A.H. Zewail, J. Chem. Phys. 69 (1978) 3350.
THEORY OF TIME-RESOLVED LEVEL ANTICROSSING EXPERIMENT R. HABERKORN, H.L. SELZLE, W. DIETZ, S.H. LIN * and E.W. SCHLAG Institute for Physical and Theoretical Chemistry, Technical University of Munich, 8046 Garching, West Germany Received 24 January, 1980; in final form 28 May, 1980
The purpose of the present paper is to present a general treatment of steady state and time-resolved level-anticrossing (LAC) experiments using the generalized master equation approach in which both radiative and nonradiative decays and dephasing processes have been considered. Several models are treated to demonstrate the theoretical method. The theory is applicable to both atoms and molecules. It is shown that under appropriate conditions, as derived in this paper, quantum beats can be observed in the time-resolved LAC experiment. Numerical calculations have been performed to show the time-dependent behavior (build-up and decay) of the time-resolved LAC experiment. These quantum beats are then a direct measurement of the microscopic coupling parameters in intersystem crossing, etc. It will be shown that combining the steady state and the time-resolved LAC experiments one can determine not only the microscopic coupling parameters but also the relaxation and/or the dephasing rate constants. Hence the particular virtues of time dependent LAC experiments are seen in these model calculations.
1 Introduction The level anticrossing (LAC) phenomenon is observed whenever two levels, say Im) and I n ) w i t h unperturbed energies E m and E n of an atom or a molecule are brought into coincidence by applying a magnetic field, provided that there exists some interaction H'mn = (m I/~' In) between Im } and In). Near the crossing point, the two eigenvalues of the total hamiltonian repel one another and the associated two eigenfunctions are mixtures o f Im } and In }. This produces an equilization of the populations of Im) and In } at the avoided crossing point if some steady state excitation mechanism produces a difference between the populations o f Im} and In }. This resonant-like variation of population can be observed by monitoring separately the intensity and/or polarization o f light emitted by Im} and In}. The i n t e r a c t i o n / 1 ' could be due to an electric field interaction, a fine or hyperfine interaction etc. Level anticrossing was observed for the first time on the sublevels of the 2 2 p manifold of Li; the coupling in this case was due to the hyperfine interaction [ 1 j. Since then a great number of other LAC experiments have been carried out for atoms and molecules [ 2 - 7 ] ~. However, all o f LAC experiments so far have been concerned 0nly with the measurement of steady state behavior. The purpose o f the present paper is to present a theoretical approach which can be applied to study not only the steady state behavior but also the dynamic behavior (i.e. time evolution) of LAC phenomena. To demonstrate the theoretical method, we consider the pseudo two-level system and pseudo four-level systems corresponding to LAC between two doublet states, and between a singlet and a triplet. Numerical calculations are performed to illustrate the effect of various parameters (e.g. energy level spacing, coupling constant H'mn, decay constants, dephasing constants, etc.) on the dynamic and steady state behavior o f the LAC phenomenon. It should be noted that although in this paper we deal only with LAC, clearly the present theoretical approach can be modified and applied to other related experiments like level crossing, double resonance, MOMRIE etc. * Permanent address: Department of Chemistry, Arizona State University, Tempe, Arizona 85281, USA. For a recent review, see ref. [ 21.
364
R. Haberkorn et al. / Theory o f time-resolved level anticrossing experiment
2. General considerations It has been shown that the time dependence of the density matrix of the system embedded in a heat bath is determined by the Liouville equation of motion [8,9] dr)/at = -i/~o~5 - i£'15- [ ' ~ ,
(2.1)
where Lo represents the Liouville operator from which the basis set is chosen, L' represents the interaction Liouville operator that causes the level anticrossing and I" represents the damping operator. In terms of matrix elements of ~, we find, for example dPnn_ dt
(2.2)
i ~ ( H , n n , P n , n - P n n ' H n ' n ) - ~ Pnn:n'n'Pn'n' h n' n'
and dpmn _ dt
(i~mn + Pmn'mn)Pmn - ~ "
'
(2.3)
(H~n'Pn', - Pmn'H~'n) .
These are usually called the master equations. Here Hnn' represents the interaction of the levels, which is responsible for the anticrossing, P describes the damping of the system and boron is the energy difference between the levels without interaction. For convenience of discussion, the properties of I~ are summarized in the following [8] !
Pnn:nn = --
(2.4)
rn'n':nn ,
f n
where Fn~nt
nn
27r ~ h %
-
~ p(b)(O)nbnb[(nno[i21,RNin,n,b)l= 6 ( E n ' n b _ nb
~""b)'
(2.5)
~(b)(0) represents the initial distribution of the density matrix of the heat bath and HRN denotes the interaction between the system and the heat bath (including both radiative and nonradiative intramolecular and intermolecular interactions), and Pmn :ran is the dephasing rate constant d) F m n :mn = ]1 ( F r o m : r a m + P n n : n n ) + P (ran:ran '
(2.6)
where P ran:ran (d) represents the so-called pure dephasing rate constant (d)
Pmn:mn
rr ~ = -h
~
nb
p(b)(O)n,bnb([(nnblICl,RNinn,b)l -- [(rnnb["HRNlmn'b)D ' 2 8(En b - En'b)
n'b
(2.7)
As can be seen from eq. (2.5), - P n ' n ':nn represents the rate constant for the transition n -+ n'. In particular, for the three-level system (a, m and n) eqs. (2.2) and (2.3) become dPnn/dt =
-
(i/h)[(H'naPan - PnaHan) ' + (HnmPmn ' - PnmHmn)] '
(2.8)
-- ['nn :nnPnn -- P n n : m m P m m -- Pnn :aaPaa , t
t
dPmn/dt = - (it°ran + Pmn :ran) Pmn -- (i/h)[(HmaPan - PmaHa,) + Hmn(Pnn - Pmm)] ,
(2.9)
respectively, where r
(.Omn
=
OOmn +
h--
I
t
(Hmm
-
t H..)
(2.10)
R. Haberkorn et al. / Theory o f time-resolved level anticrossing experiment
365
and for example Fnn:nn
= -Faa
(2.11)
: n n - - F r o m :nn •
The matrix elements of 1:" (or rate constants) are schematically shown in the following: n
~-Fnn:mm~:P_m__m!_nn__~-
7
m
I -Paa :nn
- P n n :aa i
It should be emphasized that P is well defined (i.e., can be calculated microscopically) and includes both radiative and nonradiative processes; for nonradiative processes, both intramolecular and intermolecular (if they exist) processes are included.
3. Three-level model We first consider the simple three-level system H m a
Here a represents the ground state and rn and n are the states in which the anti-level crossing (or interference) takes place. If we assume that in eqs. (2.8) and (2.9) klan = 0 ;
Ham
= O,
(3.1)
which means that there is no interaction of n and m with the ground state a, respectively, and eliminate Paa by using the relation (3.2)
Paa + Prom + Pnn = 1
we obtain the pseudo two-level problem dpmm/dt
= -
( 1" / h ) ( g m n' P n m
-
PmnHnm)'
- - F r 'o m : m m P m m
-
rmm'
:nnPnn
-
Pmm
t
dPnn/dt
= -
dPmn/dt
' -- PnmHmn)
(i/h)(H'nmPmn
= -iCOmnPm
n +
(i/h)
' Onn - P n n '. m m P m m - Fnn:aa Fnn:nn
-
H'mn(Pmm
-
Pnn)
:aa ,
(3.3) (3.4) (3.5)
and •
tt
•
(3.6)
t
dPnm/dt = -lCOnmPnrn + ( l / h ) Hnm (Pnn' -- Pmm)
where •
tl
.
t
.
IOOmn = lCOmn + Pmn :mn ; t
.
Pnn:nn =Fnn:nn --l-'nn:aa,
.
t
.
=From:ram
-
t
I'nn:mm = F n n : m m - F n n : a a ,
t
From:ram
tt
lOOnm=169nm + Pnm :nm ,
Fmm:aa
,
t
Fmm:nn
= Fmm:nn
-
Fmm:oa
•
(3.7) (3.S) (3.9)
366
R. Haberkorn et al. / Theory o f time-resolved level anticrossing experiment
In the following, we shall present our study of eqs. (3.3) - (3.6) for the time dependence of the system and the steady state behavior of the system. We first consider the solution of eqs. (3.3) - (3.6). For this purpose we shall employ the Laplace transform method. Applying the Laplace transformation to eq. (3.3) yields PPrnrn -- Prom(O) = -- ( I" / h ) . ( H m .n P n m. - .P r n n .H n m ). - I'm . : m. P m m - F m
:nPnn - P - l [ ' m :a ,
(3.10)
where Prom represents the Laplace transform of prom(t), Prom = P r o m ( p ) =
.;
e -vt
prom(t)dt
(3.11)
o
and F;n
-- r r 'n m
: Fn
t
; rn
ol
t
.
rm :n = F m m :nn ,
,
P m :a = I~rnrn :aa •
(3.12)
.
(3.13)
Eq. (3.10) can be rewritten as t
t
--
•
t
t
(p + rm :m) V m z + r m :nOn. -- (iHi, m/h) tim. + (iHmn/h) -~.m = Prom(O) - p - I r m :a . Similarly we obtain t
- -
t
Cn :rn P r o m + (P + r n
--
:n) P nn
•
t
--
.
+ ( i H n r n f f t ) Pmn •
ft
-
!
( i H m n f f t) P n m = P n n ( O ) - P - l [ ' n
- -
:a
(3.14) (3.15)
- (W'mnlh) ~,.m + (g4Lnlh) ~n. + @ + ~ , . n ) P.,n = p . , . ( o ) and .
tt
--
(3.16)
(iliUm/h) Prom - (iH~ml h) Pnn + (P + lWnrn) Prim = Pnm(O) •
From eqs. ( 3 . 1 3 ) - (3.16), we can solve forpmm, Onn, Pmn and~nm. For example prom(O) 1 ~?1"1
=
~.(p)
-
-
rm :JP
Fm :n
-iH'm/h
iHmn/h
Pnn(O) - I'n :alp
P + I~n :n
iHPnm/h
-iHmn/h
;ran(O)
iHm.l~
p + iW~nn
0
Onto (0)
-itInm/h
0
p + iWnm
(3.17)
where t
P + r m :m t
A4@) =
t
.
I-'rn:n
t
--iHnmff l t
.
t
.
t
iOmnffl .
t
Fn :m
P + Fn:n
iHnm/h
-itt'mn/h
-iHmn/h
itt'mn/h
p + iCOmn
0
iHnm/h
-iHtnm/h
0
P + iCOnm
(3.18)
It will be shown that this will lead to the equations which are commonly used in analyzing the LAC experimental data. 3.1. Case I - build-up
Here we study the build-up and steady state behavior of the system and in this case, we make the following approximations. The system is initially in the ground state a and the pumping is described by a rate constant, where state n is pumped by 1-'n:a and state rn is pumped by Fm :a- Further we assumed that there is no transition
367
R. Haberkorn et al. / Theory of time-resolved level anticrossing experiment
between n and rn other than that caused by the LAC. The decay rate of state n and m is equal to P; Pnn(O) = Pmrn(O) = Pmn(O) = ,Onto(O) = 0 ;
["n :m = rrn :n = 0 ;
Pn :n = Prn :m = [ ' .
Using these approximations, eq. (3.17) becomes Pm :a
(-1)
Prom
pn4fp)
--Prn :a
--i[tnm/pl
iHmn/h
Pn :a
P + r -- rn: a
iHnm/Pl
-'iHmnff1
0
iH'~dh
p + icO~nn
0
0
--iI-I;mffl
0
P + iWnmt
(3.19)
where P + l~ -- Pm :a
adp) =
-r. •
--I'm :a
:a
--il-l;m
p + p - r n :a
t
,
iH;,m/h
t
--iltrnn/h
iItmn/h
iHnm/h
-iHnm/h
il-Imn/h
.
.
--igm,,l~
tr
p + 1COmn
0
0
p + i~nm
(3.20)
The steady state solution of Prom (t) is given by r m :a
(-1) Prom (oo) - a4 (0)
- I ' m :a
-iI-I'nm/h
iHmn/h -iHmn/li
r n :a
F - Pn :a
iHnm/Pl
o
i~m,,Ih
ioo;;,.
0
0
--iH'nmffl
0
" "m 109n
(3.21)
Eq. (3.21) can be simplified as
Prom(°°)-
rm :a r'
(2 IHm. 1 2 / r ' h 2 ) ( r . , . / r ) ( r . :a - Fm :a) (COm.) 2 + r £ . + 4 r . , . I H m n l 2 / P h 2
'
(3.22)
where P '= F - Fn:a - Pm:a •
(3.23)
Eq. (3.22) describes the steady state population and the lineshape for coherent excitation of levels rn and n (coherent excitation here means Pnm(°°) --/=0; this is the case whenever I'n :a :~ Pm :a) and is commonly used in analyzing the experimental data of the steady state LAC experiment [2]. But as can be seen from the approximations stated above, eq. (3.22) is valid only when P n :m = F m :n = 0 and l-'n :n =Fm :m = PThe same result can be obtained by setting dPmm/ dt = dpnn/ dt =' d p m n / dt = dpnm / dt = 0 .
(3.24)
Next we shall study the time dependent behavior of Prom (t). It can be expected that in this case one would obtain the time evolution of the build-up of Pmn(t), Pmm (t) and Pnn(t). Notice that eq. (3.19) can be simplified as -ffmm
_ (--1)[Fro :a(P + P)(P + i6°mn)(P + iC°nm) + (21H'~nl2/h2)(P + I'mn)(I'm :a + rn :a)] p(p + r - I~m :a - Pn :a)[(P + F)(p + iWmn)(P + i ~ n m ) + (4 [Hmn 12/h2)(P -I" Pmn)]
(3.25)
or ( - r m :a) [(P + r ) ((p + rmn): + ( ~ m . ) : ) + (2 la~,. 1 2 / h : ~ + rmn)(1 + r . :a/rm :a)] ffmm - p(p + r - r m :a - rn :a)[(P + r ) { ( p + rmn) 2 + (ag~n} + (4lH~nnl2/h2)(P + rmn)]
(3.26)
368
R. Haberkorn et al. / Theory of time-resolved level anticrossing experiment
Here the following relation has been used •
rt
.
tt
t
(19 + ICOmn)( p + 16grim) = (]9 + r m n ) 2 + (6)mn)
2
(3.27)
.
If 1" = Pmn, then eq. (3.26) reduces to Finn) 2 + (LOmn)2 + (21H'mnl2/h2)( 1 + Ian:a/Fro : a ) ] p(p + I" -- I~m :a -- I~n :a)[(P + I~mn) 2 + (O)mn) 2 + 4Ill'ran 12/h2]
(--Fm
-Prom
:a)[(P
+
or
r
Prom -
(--I'm :a)[(P + Pmn) 2 + (6°mn)
2
+ (2]Hmnl2/h2)( 1 + P , : a / r m :a)l
p(p - x,)fp
- a2)(p
- x3)
(3.28)
,
(3.29)
where X3 = - ( P - P m :a - Pn :a) and )t~ and X: represents the two roots of the equation t
2
t
(p + Finn) 2 + (¢.Omn) +4[Hmnl2/h 2 = 0 .
(3.30)
Carrying out the inverse Laplace transformation of eq. (3.29), we obtain Pmm(t) (--Pm
:a)
-
' 2 (_Pn:a _ 1 ) [ Pmm(°°) +2[Hmnl ~-m
:~-
h2
\ I ~ m :a
exit
e•2t
)klCA' - - ) k 2 ) 0 k l - - X 3 ) ~ )k2CA2 -
1
) k l ) 0 k 2 - - )k3)
eXat[()~ 3 + Finn)2 + (COm.) , 2 + (21H~nlE/h2)(1 + rn:a/rm :,)l +
)k3 (~k3 _ )k 1 )()k 3 _ ~k2 )
,
( 3 . 3 1)
where Pmm (oo) represents the steady state population of Prom(t), and is given by eq. (3.22). In other words, eq. (3.31) describes how the LAC signal reaches the steady state. The terms involving e M t and e M t give rise to an oscillatory behaviour of the build-up of the system.
3.2. Case H - decay Here we shall study the decay behavior of the system. For this purpose, we shall assume an arbitrary population and phase of the levels n and m at the beginning, that is,
Pmn(O):/:O,Pnm(O)#:O;
Pnn(O)eO, Pmm(O):/:O;
Pm:a=0, rn:a=O.
Experimentally this means that we have a coherent population at the beginning with the excitation source removed. In this case we have --
t
2
P
P nn = A4 1 (P)~0mm(0)[(P + I'n:n) [(P + ['mn) 2 + @Omn) ] + (21Hmnl2/h2)(P + Pmn)]
+p,,(O)[(21Hm, 12/h2)(P + rmn) - Pm:n[O + Fro,) 2 + (COm,)2 ] ] + Pmn(O)(iHnm/h)(P + r m :n + r n :n)(P + i6°nm + r m n ) + Pnm(O)(-iHmn/h)
X (p + rm :n + r . :n)(P + iCO~n + Finn)) ,
(3.32)
where r
2
A,O,) = [0, + r m . ) = + ((-Omn) ] [(l19 + I~m :m)(P + rn:n) -- rm :nrn:m]
+ (2[HmnlZ/h2)(P + Pmn)(2P - Pa:m -- Pa:n) •
(3.33)
Next we consider some particular cases of eq. (3.32). For example, if Pn :m and Pm :n are negligible and Prnn = Pn :n = Pm:m (no pure dephasing), which means that the transition between the states n and m is only due to LAC and that both states have equal decay rates, then eq. (3.32) reduces to Pmm(P) =-pmm(p)l + Pmm(P)2 ,
(3.33a)
R. Haberkorn et aL / Theory of rime-resolved level antierossing experiment
369
where Pmm(P)l =
Pmm(O)[(P + Pmn) 2 + (C~mn) 2 + (21Hmn 12/h2)] + (2triton
I=/h2)p,m(0)
(3.34)
(P + Pmn)[(P + Pmn) 2 + (~°m,) 2 + 41H'mnl2/h21
and (llfl)[Pmn(O) Hnm( p + i~3'nm + I'mn )
' Pnm(O)Hmn(P +"lCOrnn , + rmn)] ,2 , (1° + r m n ) [ ( P + l~mn) 2 + (6Omn) + 41H'mnl2/h2]
•
Prom(P)2 -
,
_
(3.35)
Notice that-Prom (P)l and Prom (P)2 are due to incoherent and coherent initial population, respectively. It follows that (3.36)
Pmrn(t) = Pmrn(t)l + prom(t)2 ,
whereA2m,= (corn,,) ' 2 + 41H'mnl2/h 2 , r 41Hmn I2 e - r m n t sin 2 ~Amn t (bAron) 2
e-I'mn t Pmm(t)x = Prom(O) -(hArnn)2 --[(hCOmn) 2 + 4 1 g m' , I 2 cos2½Amn t]
+Pnn(O)
(3.37)
and 4COmne-Pmn t Prom (02 . . . . . . . . . . . . Re [Omn(O) Hnm I sinZ ~ Amnt hA2m,
2 e - rmn t
t
Im [ p m , ( 0 ) H , , ,
hAm,
]
•
(3.38)
Pnn(t) can be obtained from the expression of Prom (t) by interchanging indexes n and m. The LAC system will exhibit oscillatory decay with the beating frequency given by Amn. Notice that if both levels have equal initial populations [Pmm (0) = Onn(O)J, then eq. (3.39) becomes
Prom (01 = Prom (0) e x p ( - P m n t ) .
(3.39)
That is, in this case, the oscillation in Prnm(t)l disappears. But if Pmn(O) ¢ 0, which relates to an initial phase relation between the two levels due to a coherent excitation, the beat can still take place through Prom (t)2. In particular, if ann(O) = 0, and Pmn(O) = 0, then eq. (3.36) reduces to Pmm(t)/Pmm(O ) = ( e - r m n t / A ~ n ) [(Cdmn , ) 2 + (41Hmnl2/h , 2) cos2~Amnt] . t
(3.40)
2
If(cOmn) < < 41H'mn IZ/h 2, then eq. (3.40) reduces to Pmn (t)/Pmm (0) = exp(--Fmn t) COS21~Amn t ,
(3.41 )
where Amn ~ 21Hmnl/h. This represents the case for the LAC experiment near the crossing point. The decay will be nearly 100% modulated and the beat frequency will be the direct measurement of the interaction strength. On the other hand, if t2 a~mn >> 41H'mn 12/h2, then (3.42)
Pmm (t)/Pmm (0) = exp(--I" m n t ) .
That is, the experiment is far from the crossing point and the oscillation term is negligible in this case. 3.3. Numerical calculation For convenience, we introduce the dimensionless quantities *
*
t = (Fn:n/I'n:n) t ,
¢
,
i
*
(Hmn) =Hmn(Fn:nlI'n:n) ,
t
,
r
,
(Oanm) = ~ O n m ( r , , : , / r , : , )
etc .
In figs. 1 and 2 we show the results of the time-dependent behavior of the build-up and the decay in the LAC
R. Haberkorn et al. / Theory o f t i m e - r e s o l v e d
370
• 5 -i
antierossing experiment
:MRG [RHO]MN
z G~,
.5
IMBG
.5
RERL [RHO)MN
[BHO]MN
W ----2--222:::: . . . .
C-)
CO
level
co:: : .
. . . . . . . . . .
- -
:33 ~2 CD
.5
RERL
[RHO}MN
FTI rCo
~ ~.~ 212 2E cD
ED 7~ Z .5
--7
[RHO}NN
2I) ZI2 ~
CD Z 7
-.5.
(RHO]NN
1.0
.5
,1
(RHO]MM
~
.3 ,
1.0_
[RHO]MM
_
22g
.1
0.0
.... ..:L I
~.~
,5
........
I 1.~
I 1.5
I 2.0
I 2.5
0.0 0.~
3.1~
.5
(r,,: a, COnm
= =
1): 5.0.
evolution of LAC experiment, build-up case .... Oan*rn = 0.5, - - Wnrn = 1 . 0 , - -
1.5
2.0
2.5
3.0
TIME
TIME Fig. 1. Time
1.0
F i g . 2. T i m e e v o l u t i o n
(Onn(O)
=
of LAC experiment,
decay case
1): .... Wn*m= 0.5, - - - - oan*rn= 1 . 0 , -
eOn*m = 5.0.
experiments, respectively, for the case Pn:n = Fm :n = 1 , (COnrn) '(H'mn) I, and Fm :n = Fn :m = 0. Notice that for the build-up case we have assumed that P~, :a = 0 and Fn :a 4: 0, i.e. only the n state is pumped. In other words, it is assumed that the transition a -+ m is forbidden. The purpose of figs. 1 and 2 is to show the effect of the decay rate constant on the time-dependent behaviors of the LAC system. Numerical calculation for many other cases has also been performed, but the qualitative behaviors are quite similar to those shown in figs. 1 and 2. In figs. 1 and 2, we show not only the time evolution of the populations Prom (t) and Pnn(t) but also the time evolution of the Re [Pmn(t)] and lm [Pmn(t)], the phases of the system. As can be seen from these figures, the quantum beat becomes pronounced when Icomnl/Fn: n > [(or IHmnl/i'n:n > 1). When we want to classify the LAC experiment we have to consider three cases, which are related to the magnitude of the damping Fn :n of-the system compared to the interaction leading to LAC. In figs. 3 a - 3 c , we show the time-dependent behavior of Pnn and Prnrn for the three cases, IHmn[ > I'n :n' IHmnl = Pn :n and IHmnl < Pn :n" Here for simplicity we have assumed f m :m = 0, I"m :n = Pn :m = 0. This means the system decays only via Pnn and there is no pure dephasing of the system. The calculation is performed for the decay of the system, and only state n is populated at the beginning. The time-dependent behaviors of Pnn and Omrn for Pn :m :/= 0, I'm :n =/=0, and I'm :m 4:0 are qualitatively similar to those shown in figs. 3 a - 3 c . Besides the excitation process the experimentalist can adjust the external field strength which will result in the energy gap change. Thus in figs. 3 a - 3 c we also show the effect of changing the energy gap (i.e., the scanning of the magnetic field in the conven=
R. Haberkorn et al. / Theory o f time-resolved level anticrossing experiment
,.o_
{RHO]NN
"'\X " " " " ,
Z
"-'\
(RHO]iNN
1.0_ //\
•
371
..
o[/X/X/;'4.'~ ] 1
0.0
.5
I
''
l.~
[
' ~'
1
L
', 1 "\
2.El
,.5
2.5
" 3.
TINE
C
ioL
[RHO)NN
5
[RHODMM
1.0_
[RNO]MM ZZI3
(D
.5
/ \, O.fl
0.0
.5
0.0
,.0
,.5
2.0
~.~
~.o
.5
O.t~
1.(3
TIME Fig. 3. T i m e e v o l u t i o n o f L A C e x p e r i m e n t , 5.0,
rnn
=
1.5
2.~
2.5
3.2
TIME = 5 . 0 ) : ( a ) Fmm * = 0 . 0 , Fnn * =1.0;.. . . Wnm '* =0.0,--tOnm '* = ' * --- 0.0, -- -- -- tOnm '* = 5.0, ' * - 2 5 . 0 . ( c ) Fmm tOnm tOnm * = 0.0, ~nm = 25.0.
d e c a y c a s e ( H n ,' *
* - 5.0;.... t o 'n* , - 2 5 . 0 . ( b ) r m* m = 0 . 0 , I'nn 1 0 . 0 , . . . . t°nm = 0 . 0 , - - - - t ° n . = 5 . 0 ,
tional LAC experiment) on the density matrix elements. As can be seen from these figures, as the energy gap increases, the quantum beat becomes less pronounced. 3.4. Further discussion o f the steady state case
Notice that in obtaining the steady state solution given by eq. (3.22) two assumptions are made, i.e. rm :n = = 0 and r m : m = rn:n = r. Without these assumptions the steady state solutions of eqs. (3.3) - (3.6) are given by rn:m
t
r-:a(Zmn P,m
(Zmn - rm:m)(amnr. :a(Amn
Pnn
- r n : n ) + r n :a(Amn + r n : n )
-----
r
rn:n)-
(Am. _ rm :m)(Am. -
(Ann + rm:n)(Am.
r ; . :m) + r .
: a ( A m n
? t r.:n) - (Amn+ r m
r~ :m) :n)(Amn +
,
,
(3.43)
+
(3.44)
372
R. Haberkorn et al. / Theory o f time-resolved level antierossing experiment
and t
.
t
Prnn
= (i/h) H r n n ( P m r n -- Pnn)/(l~Ornn + P m n ) ,
(3.45)
Amn
= -2I'mnlH~12/[(hcO'~n)
(3.46)
where 2 + ( h F m n ) 2] •
In other words, eqs. (3.43) - (3.45) give us the exact results of the steady state population and the phases of the three-level LAC system. Other particular cases can be deduced from these equations. t t f f It should be noted that in some cases Hna (or Han ) a n d H m a (or Ham) may not be zero, but the effect of these terms is negligible as long as the energy gaps COna and COma are much larger than I H a m 1, I H a n l , and I H n m l (see appendix).
4. Fivedevel model 4.1. Case I
Next we consider the following five-level system which represents the case for the anticrossing of two doublets connected with the same ground state: (D~)
.n'
(D1)
n
(D2)
m
(D~)
m'
(So)
a
n and m are the levels in which the anti-level crossing takes place. In other words, we have only/~/~nn ~ 0 and the other matrix elements o f / ~ ' are zero. In this case, from eqs. (2.2) and (2.3), we find dpmrn/dt
= -
" ) ( n m.n P n m (l/h
-- . P m n H n m ). - P m
. ' m ' -- P m :n Pnn -- P m : n ' P n ' n ' -- P m :a , :.m P m m -- . F m : m ' P m
(4.1) dPnn/dt
= - ( l"/ h ) ( H n m ' Prnn - P n m H m' n )
-
P n. ; n P n n .- r n :n'. P n ' n ' --. P n :m Omm -- F n : m ' P m ' m
' -- Pn :a ,
(4.2)
= - (it°ran +
dprnn/dt dOn'n'/dt dpm'm'/dt
-- (i/h)Hmn(Pnn - P m m ) ,
Pmn)Pmn
r
(4.3)
t
= -Pn'=n'Pn'n'
- rn':nPnn
= - - I ~ m' ' = m ' P m ' m
- P~':m Prom
' -- ~ m' ' : m P m m
-
-
rn':m'Pm'm'
-
rn':a ,
-- F m' ' : n P n n -- F m ' : n ' ton ' n ' -- P m ' : a
(4.4) (4.5)
by eliminating Paa and an expression similar to eq. (4.3) can be written down for O h m . Let us first consider the steady state case. The steady state solution for the pseudo four-level system is given by 1
Prom(°°) =~
I ['m:a' Pn :a
-(Amn+Pm:n+P~n:m'Rm'n+P~n:n'Rn'n) Am.
- r; :. - r;:~'Rm'n
I
(4.6)
- r;:n'Rn'n
and !
1
Pnn(°°)
=~)-2
Amn
-I'm:m
-(Amn
-Pm:m'Rm'm
+ Pn :m + r;, : m ' R m ' m
-Pm:n'Rn'm
+
r;,:.'R.'m)
Pm:a
r . :a '
(4.7)
R. Haberkorn et al. / Theory o f time-resolved level antierossing experiment
373
where t
t
¢
-(Amn
-(Amn + rm:n + r~, :..'Rm'n + rm :n'Rn'n) I
t
Arnn - F m : r n - F m : m ' R m ' m
D2 =
- Fm:n'Rn'm
F
t
A mn - ['n:n -- r n :m ' R m 'n - r n : n ' R n ' n
+ Fn: m + F n : r n ' R m ' m + F n : n ' R n ' m )
t '
(4.8) --1
Rm, m = A 2
r
t
t
(Fm':n'I'n':m
t
-- F n , : n , P m , : m ) ,
(4.9)
n :n I r a ' : n ) ,
(4.10)
Rm'n = A~l(pm':n'l~n':n
(4.11)
R n , m = A ~ - l f p~.--n ' , :m ,r, r , :m - - ~ m' ' : m ' F n ' ,: m ) , --in t
t
t
t
R n ' n = A { 1 ( F n ' : m '['rn':n -- F m ':m ' I ' n ' : n )
(4.12)
A.,n = - 2 r m n IHmn 12lh [(~mn) ~ + r ~ n ] ,
(4.13)
and
t
!
t
with A= = P~n' :m'l~n ' :n' -- Pro' :n'Fn ' :m'. It should be noted that in the literature I~,n :m' ~ F m : m ' , I"rn :m ":" Pm :m, Fn :n ~ l?n :n' etc. have been assumed. That is, the contributions from the pumping rates r m :a, r n :a etc. have been ignored [cf. eqs. (3.8) and (3.9)]. Derouard et al. [10] have also studied this system, but the following approximations have been made by them: (a) no cross-relaxation between different doublets t
!
~.
t
. . m =. r n . . m., = F n. , = m = . Fn,:m, = Fm,:n . FIn:m =I~n:rn ' =. F m : . n = F .r o : n = F r o : n' = I'n.
t
= F m ' : n ' = 0 " (4.14)
(b) equal lifetime of the states, i.e. l~m :m = I~m :m = I"n :n = rn :n = F n' , : n , = F m, , : m , = - F o - F l ;
(4.15)
(c) equal relaxation rates within the doublets t
rn:n'
t
= rn :n' = rm :m' = Fm'=m
= rl
•
(4.16)
Using these approximations, eq. (4.6) reduces to
rm :~ Prom -
~t
(rm +
:a --
rn :a)
21a~nnl 2 ,
"Y
(h¢Omn)
2
+ (hPmn) , + (4Pmn/m)lHmnl2
,
(4.17)
where 7 = - P o (Yo + 2P1 )/(Co + P x), which is equivalent to the expression derived by Derouard et al. if we make the identifications, Po = _~/(o) and P1 = - ~ ( ' r (1) - 7(0)). 4.2. Case 11
Another five-level model to be considered is
(Sl)
n
(T+ 1)
m
(To)
m'
(T_I)
m"
(So)
a
This corresponds to the anti-crossing between the singlet state and the triplet state. In this case, the master equa-
374
R. tfaberkorn et al. / Theory o f time.resolved level antierossing experiment
tions for the pseudo fourqevel system are given by d p n n / d t = -(1h/)(HnmPmn"
'
- PnmHmn)'
- P n. : n P n n -. P n : m .Prom - .P n : m ' P m ' m '
- Pn:m"Pm"m"
- Pn:a ,
(4.18) dPm m / d t = - ( i / h ) ( H m n P n m
- P m n H n-m )'
rm' : m P m m
-
.Pnn
rm:'
rm'
-
:m
'Pro ' m '
•
t -- Fm :m "Pro "m"
-
dPmn/dt = -(i~mn
(4.19)
P m :a ,
+ P m n ) P m n - (i/h) H m n ( P n n - P r o m ) ,
?
!
(4.20)
t
p
4.21)
d P m ' m ' / d l = - - I ~ m ' : m ' P m ' m ' -- I ~ m ' : m " P m " m '' -- F m , : m P m m -- F m ' : n P n n -- F m ' : a
and .
n
~- _
dPm m / d t
?
.
Fm
.
.
.
:m P m m
t
t
?
- I~m":m'Pm'm ' - r m " : m P m m
(4.22)
- ['m":nPnn - I'm":a •
Let us first consider the steady state case. F o r the case Fro' :a = 0, Fm" :a = 0, we find 7 Fm "a •
? : n + F mp : m ' R m ?' n + ['m
-(Amn
Prom = ~ - ,
1J2 [ I'n :a
Amn - rn
:n
+ F mt : m " R mr" n )
, , - Pn : m ' R ' m ' n - r n : m " R m " n
(4.23)
and ?
= 1 Pnn
t
Amn-Fm:
-~2
F
t
r
m -Pm:m,Rm,m-['m:m,,Rm,,
- ( A m n + I ' n : m. + V n.: m ' R.m ' m +. I ' n : m
m
F m :a rn:a
"R~n"m )
(4.24)
,
where
_(Amn - r m : ~
D' 2
--
- P~n:m'Rm'm
F
]-(Amn
-1-'~:m"R~",n
t
F
+ Fn:m + Pn:m'Rm'm
-(Amn
t
l
+ ?m:n t
+ rm :m'Rm'n t i
Amn - Pn'n - Fn:m'Rm'n
+ Fn:m"Rm"m)
+ rm:m"Rm t ¢
"n)
- I"n:m"Rm"n
(4.25) F
Rm'm
= (1/A~
t
.
)(rm
':m"
.
.
t
t
~
(4.26)
I"m,:m ) r
Rm'n = (1/A2)(rm ':m"r~" I
- -.I " m , , : m , ,
t
¢
:.
- Vm" :m " F i n ' : n ) ,
t
?
(4.27)
t
(4.28)
R m , , m = ( 1 / A 2 ) ( P m , , .. r e , I ' m , . , m -- F m , : m , F m , , : m ) t
t
I
t
f
t
:n - F m ' : m ' F m " :n)
Rm"n = (1/A2)(Fm" :~'r,,'
(4.29)
and Pm':m'
Pm":m
"-
Pm" .mIFrnr.m ,
(4.3o)
.
Notice that r
Pm'm' = Rm'mPmm
(4.31)
+ Rm'nPnn
and f
r
Pm"m" = Rm"mPmm
.
(4.32)
+ R m nPnn •
In particular, if one uses the a p p r o x i m a t i o n s [see eqs. (4.14) - (4.16)] t
t
?
r~:m = r ; . : n = r L ' : n = r n : m ' = r n : m " = r m " : n -- 0 , , Fm :m' = I'm':m
= F m p :m
,,
= F r ro " :m = F mr ' : m
"
=
r m "
• m'
(4.33) =
r
1
(4.34)
R. Haberkorn et aL/ Theory of time-resolved level anticrossing experiment
375
and P;n: m = ['m':m' ' ' ' --=- P o - Pl , = 1"m":rn" = Pn:n
(4.35)
then the steady state solution can be simplified as Amn(1"m :a + 1"n :a) + 1"m :a(Co + 1"1) '°ram = 2Amn(ro + I"1 - 1"]/Po) + (Po + r l ) ( F o + Pl - 2F]/Po) Pnn -
Amn A m + Fo + F1
Pmrn +
Pn:a A m n + 1"o + 1"1
'
(4.36) (4.37)
,
Pm 'm' = Pm"m" = (1"1/1"o) Pmm •
(4.38)
This gives us the linewidth and the steady state population of the system. The time-dependent behaviors of the above two pseudo four-level systems can be studied by solving respectively eqs. (4.1)-(4.5) and (4.18)-(4.22) by the Laplace transform method. We have also performed the numerical calculation to study the time dependent behaviors of these two models. Since the results are qualitatively similar to those of the pseudo two-level system, they will not be presented here. Although the steady state and time-resolved LAC experiments can be qualitatively understood by using the three-level model, for the purpose of quantitatively analyzing the LAC experimental results to determine the LAC experimental results to determine the LAC coupling constant, decay rate constants and dephasing constants, it is necessary to treat the physically more realistic models like those presented in this section.
5. Discussion
In the previous sections we have presented a theoretical approach for the systematic treatment of steady state and time-resolved LAC experiments. It should be noted that in deriving eq. (2.1) which is our starting equation, the Markov approximation has been made, the singular perturbation method has been used to second order and the system and the heat bath are assumed to be decoupled initially. In the time range where the Markov approximation is not valid (i.e., the transient region) oscillatory decay may be observed depending on the nature of the memory function involved in the relaxation processes. However, the quantum beat discussed in this paper is mainly due to the interference between the LAC states and not due to the non-markovian effect. Conventional LAC experiments are carried out under steady state conditions; in this paper, we have treated the time-resolved LAC experiment. In this connection, it is often convenient to work in the frequency domain rather than in the time domain. In that case, we need only to carry out the Fourier transformation of the results for prom(t), Pmn(t) etc. obtained in the previous sections. For example, the Fourier transform of Prom(t) given by eq. (3.40) can be expressed as
Prom((0) prom(O)
-
2P,..
'
(0mn)
Am n2
2 +21H,,ml 7 1 IHmnl r + ~ J (02 + F2n ~ L ( ( 0
1
+ mmn)
2 +
1"ran
+
1
((0 - mmn) 2 + r2mn
(5.1) which consists of the superposition of lorentzians. We have shown that the quantum beat can be observed in the time-resolved LAC experiments under suitable conditions. From the pseudo two-level model, we can see that favor able conditions for observing quantum beats are [see for example, eq. (3.40)] 2
(5.2)
Amn >~ PZmn
(the dephasing rate constant is smaller or comparable to the beat frequency) and r
41Hmnl
2
i
~ h((0mn)
2
(5.3)
R. Haberkorn et al. / Theory of time-resolved level anticrossing experiment
376
c'*= 2H"
II
t
2H* Fig. 4. Quantum beat conditions.
(the LAC interaction is comparable or larger than the energy spacing of the unperturbed levels). These conditions are shown in fig. 4. In this figure, we have plotted ¢Omn/I~rnn against 21H'mnI/hFmn. The favorable conditions for observing quantum beats are shown in region I surrounded by the heavy lines. Notice that in making the plot of fig. 4, it has been assumed that Pn :n = I'm :m" To see the effect of Pn :n 4: Pm :m on the observation of the quantum beat, we consider the case of COmn = 0. In this case, eq. (3.17) for the decay of the system after removing the excitation source can be written as ,
Prom = ~°mm(0)[2h -2 Inmn I + (p + rn :n)(P + rm :m)] t
2
t
+ 2h -2 Inmnl Pnn(O) + 2(e + Fn :n) Im[HmnPnm(O)] } , 12 - ~(Fm :m - Pn:n)Z]}-i × {(P + Fmn)[(P + Finn) z + 4h -2 Inmn
(5.4)
Here it has been assumed that Pm :n = Pn "m = 0 and r(d) = 0; i.e., the pure dephasing is also zero. Carrying out • Iron the inverse Laplace transformation of eq. (5.4) yields
Pmm(t) = Pmm(O) e-rmnt~L2h2fm nH l ~nnzl+ (1
~]coslHmnlZ]2~mn t + (Fn2~mn:n--
Finn)
sin
2~mn t]
+ Inmnl~ Pnn(O) e-trent(1 - cos 2~mnt) + 2 Im[(Hmn/h) Pnm(O)] e -rmnt 2h 2fmn 2~mn sin 2~mn t ,
(5.5)
where ~mn 2 = h - a IHmn , I2 - ~ ( F n :~ - Finn) 2 • Eqs. (5.4) and (5.5) indicate that the quantum beat disappears if /3:mn < 0 or (rn:n -- rmn)2/16 > IH'mnl2/h2. The theoretical approach for LAC presented in this paper takes into account not only the decay rates due to radiative process, radiationless transitions and collisional processes, but also the dephasing rates due to inelastic and elastic (or pure dephasing) processes. It should be noted that in calculating 1a to the second order approximation with respect to HRrq, the coupling between the system and the heat bath, the collisional rate constant obtained from expressing the matrix elements of P in terms of the concentration of the collision partner corresponds to that obtained by using the collision theory in the first Born approximation [11,12]. Recently, quantum beats have been reported by Chaiken et al. [13] for biacetyl in the gas phase. We may qualitatively interpret their results by using the pseudo two level model [see for example eq. (3.40) and figs. 3 a - 3 c ] . l* r t . . In this case, we have IHmnl > Pn :n'mmn >~>Finn and h6onm < IHmnl; I.e., the luminescence decay is much slower than the oscillation (14). In the theoretical treatment of the steady state and time-resolved LAC experiments presented in the previous
R. Haberkorn et al. / Theory o f time-resolved level antierossing experiment
~.a]
377
POPULRTION OF STRTE N
233
5.5
L/_.-
@.
- .......... I
1.@
]
I
2.0
3.@
- . . . . . . . . . . . . . . . . . . . . . . . . . . . . I
I
~.~
5.@
I
6,1~
I
7.1?
I
8.8
9.
TIME
Fig. 5. Effect of incident light on LAC experiment. The light intensity is expressed in terms of the Rabi frequency I~E;From = O, rnn = 1, Hmn = 1, COmn - 1; ... is.E= 0.6, - - - ~LE= 2.0, ~E = 6.0.
sections, the pumping o f the system has been described by the rate constants like Pnn :aa, r m m :a~, etc. Suppose that the pumping in the LAC experiment is carried out by radiation. In this case, in addition to H ' involved in eqs. (2.1)-(2.3), we need the interaction between the molecule and radiation which in the dipole approximation can be expressed as (5.6)
[1; = - It" E ( t) ,
where It represents the dipole operator and E ( t ) is the time-dependent electric field vector of the incident radiation. Here we have used the semiclassical theory for/lr- Using eq. (5.6) one can study the effect of light intensity, excitation duration, excitation bandwidth, Doppler broadening etc. on the LAC experiment. For example, in fig. 5 we show the effect of light intensity (which is expressed in terms of the Rabi frequency/.rE) on the build-up of the populations in the LAC experiment for the case Itma 0 and Itna ~s 0 (i.e., only the n-state is dipole-allowed). As can be seen from fig. 5, the time-dependent behaviors of the system are sensitive to the incident light intensity. It should be noted that if the time scale is in microseconds, the unit of the Rabi frequency is MHz and the value of 6 corresponds to dye laser pumped by a cw argon ion laser [15]. It is important to note that in the steady state LAC measurement one can only obtain the position and width of the spectra [see, for example, eq. (3.22)], while in the time-resolved LAC measurement one can obtain information like beat frequency, relative decay (and/or build up) amplitude and decay constant (see for example figs. 1 - 3 ) by changing the energy gap (sc~inning the magnetic field strength). In other words, whenever possible, it is useful to t carry out both steady state and time-resolved LAC experiments. In this way, one can determine not only H m n but also rate constants I~mm :ram, From :nn etc. and dephasing constants Finn :ran etc. =
Acknowledgement Financial support of the Alexander-von-Humboldt-Foundation, the NATO Scientific Affairs Division and the Deutsche Forschungsgemeinschaft is gratefully acknowledged.
378
R. Haberkorn et al. / Theory o f time.resolved level anticrossing experiment
Appendix t
In this appendix, we shall study the effect of nonvanishing H~n, Hna on the steady state solution of the pseudo two-level system. For this purpose, the perturbation method will be employed, •
t
(1/h)[X(HnaPa n
t
-
-
!
•
F
r
t
t
!
r
+ r . : . p . . + Fn:mPmm + I'n ."a = 0 ,
PnaHan) + (HnmPm n -- P n m H r n n ) ] t
t
(1/h)[X(HmaPam - PmaH'am) + (Hmn Pnm - PmnHnm )] + r m :mPmm + r m .., P.. + rm .-a = 0 , t
(iCOmn + Finn) amn + ( i / h ) [X(HmaPa n - PmaHan ) + Hmn(ann - P m m ) ] = 0 , (~lOma" '
+ V m a ) P m a + (i/lT)[XHrna(Paa'
(iCO~a + F . a ) P n .
t
(A.4)
t
+ (i/h)[Xl-Ioa(P~a -- P n . ) + n n m P m a
-- )kPnrnnma]
(A.2) (A.3)
- XPmnH~a] = 0 ,
- P r o m ) + HmnPna'
t
(A.1)
t
(m.5)
= 0,
where X represents the perturbation parameter. Pnn, P m m , P m n , Pan and Pare have to be expanded in power series of X; for example for Pnn, we have = t~ r l (F/yl 0 ) "1" )'t~(1) ' "t" n r / + ~k2
Pn/2
p(2n) +
(A.6)
....
In the zeroth order approximation, we find p(O) _ _ (o) _ 0
net - Pma -
(A.7)
,
and t.,nn^(°),p(Om) and Pmn(°) are given in section 3. In the first order approximation, we find p(1) - n(l) (1) nn - ~'rnm = Pmn = 0 ,
(iCOma+ •
'
(A.8)
i, ma)p(lm) + (l/h) • [nmaCOaa ' (0) -- H.(0) m m . ) + n m n P n a(1) -- p(mO~/~a] = 0
(a.9)
'
and
(ieO~a + r.a)p~ )
+ (i/h)[H~a(p~) -
-
,,(o)): + /"4' n, m t ~n(1) V'nn m a _ en(o)/4, nm"ma] ] = 0
(A.10)
•
Eqs. (A.9) and (A.10) can easily be solved
n(1)=g;/h~rH' (n(o) r~na ~1 : t na~'aa
-
-
^(o)): - HmaPnm]/(lCOna ' (o) " ' + Pna) t~nn
(A.11)
and Pma ,.,(0) 4 ' n(O) 1/(i,.,' + P m a ) • (1) = (i/h )[_H~na(p(Oa~) _ ~" m r n ,~ + /"'na~" mn,tk'wma
(A.12)
Next we consider the second order approximation, • t (1) fit r~(2) (1/h)[(HmaPa m - p(lm)aHam) + [v'rnnt-'nm
n(2)/4t
~] " [ ' I ~ t
t'mn"nmzJ
(i/h )[(HmaP(al)rn - p(ml)aH'am) + (/4'v"mn~" nrn~(2)_ ~'rnn--nm'/~4(2)'
n(2)+r'
nlm-'nn
r~(2)
--n:mt~mm
=0
,
2) +pt ~(2) = 0 :J3l + I ' m, :m p (m m - m :nr'nn
(A.13) (A.14)
and • '
OCOm
n
(2). + (i/h)[fH~naP(aln)- p(')H'ma__an)+/4'--mn~s'nn(n(2)- p(m2L)] = 0 . + r m . )-P m
(A.15)
F r o m eqs. ( A . 1 1 ) and ( A . 1 5 ) , w e can see t h a t the c o r r e c t i o n terms P~)n, Prnm-(2)and Pmn(2)depend on Pna(1)and PmaO) ¢ t which in turn depend on the energy gaps COnaand ~)ma. In other words, if the energy gaps are large compared with the magnitude of H~n and H~n, the contribution to [ n n , Prom and P m n from H~n and H~m is small (i.e., the second t t order with respect to Han and H£~n). Explicitly t"nn~ A2) Pmrn ^(2) and P^(2) m n can be found from eqs. (A.13), (A.14) and (A.15),
p(2) mm = { ( B m n -- P m : n ) [ C m n + 2h -1 Im(P(n~t/an)] - (Bran + r n': n ) ( C m n x
[(Sm.- r ; ,
:n)(Bmn - P n : m ) - (Bmn + r n : n ) ( B m n +
rm : m ) ]
-1 ,
-- 2h -1 Im(P(ml)H'am)} (A.16)
R. Haberkorn et aL / Theory of time-resolved level anticrossing experiment
379
p
t
r
× [(B,,, - V, :m)(Bm, - I'm : , ) - (Bin, + I', :,)(Bin, + I'm :m)]-t
(A.17)
and O(m2)ncan be obtained from substituting eqs. (A.16) and (A.17) into eq. (A.15). Here Bran and Cmn are defined by Bmn = 2I~mn IHmn 12/[(ho3rnn) 2 "t" (h Fmrt) 2 ]
(A.18)
Cmn =2h-2 Re [Hnm , , ( 1 ) -- PmaHan)/(lO3mn (1) , • ' + I'mn)] (amaPan
(A.19)
and
References [ll T.G. Eck, L.L. Foldy and H. Wieder, Phys. Rev. Letters 10 (1963) 239. [2] H.J. Beyer and H. Kleinpoppen, in: Progress in atomic spectroscopy, eds. W. Hanle and H. Kleinpoppen (Plenum Press, New York, 1978), Part A, pp. 607-637. [3] V. Macho, J.P. Colpa and D. Stehlike, Chem. Phys. 44 (1979) 113. [4] L. Benthem, J.H. Lichtenbelt and D.A. Wiersma, Chem. Phys. 29 (1978) 367. [5] J.A. Mucha and P.W. Pratt, J. Chem. Phys. 66 (1977) 5356. [6] D.H. Levy, J. Chem. Phys. 56 (1972) 5493. [7] T.A. Miller, J. Chem. Phys. 58 (1973) 2358. [8] S.H. Lin and H. Eyring, Proc. Natl. Acad. Sci. US. 74 (1977) 3623. [9] R.I. Cukier and J.M. Deutch, J. Chem. Phys. 50 (1969) 36. [10] J. Derouard, R. Jost and M. Lombardi, J. Phys. Letters 37 (1976) L135. [11] R.D. Levine, Quantum mechanics of molecular rate processes (Oxford Press, London, 1969) pp. 70-78. [12] M.L. Goldberger and K.M. Watson, Collision theory (Wiley-Interscience, New York, 1964) pp. 344-383. [13] J. Chaiken, T. Benson, M. Gurnick and J.D. McDonald, Chem. Phys. Letters 61 (1979) 195. [14] Detailed analysis to be published. [151 K.E. Jones, A. Nichols, and A.H. Zewail, J. Chem. Phys. 69 (1978) 3350.