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Transition probabilities for coherent multiphoton absorption processes
Chemical Physics 12 (19761 291-295 @ North-Holland
Publishing
TRANSlTtON
Company
PROBABILITIES
R’eccived 4 September
FOR COHERENT
MULTIPHOTON
ABSORPTION
1975
The feusibility of coherent multiphoron propn_=tion effects soch as two-photon self-induced by calcularinp the coherent trnnsirion probabilities for a multiphoton process. For a two-photon system, periodic probability functions are obtained which increusc smoothly as the intermediate with the radiation field. The results for a coherent multiphoton excitation of a multilevel system “strong signal theory” Tar a one-photon excikrion of 3 two-level system.
1. Introduction
and short
interaction system.
transparency is examined excitation of a three-level state approaches resonance arc an extension of RPbi’s
sections based on the “golden rule” [I I] It is obvious that there are experimental conditions in which multiphoton consecutive colzcrer~~ transitions should be taken into account. Moreover, the coherence time (the dephasing time T2) of a complex system may be very short and even comparable to the uncertainty in time during the interaction with the radiation field. Thus, as will be shown beIow, no cIear distinction can be made in such cases between consecutive and simultaneous multiphoton transitions. Simultaneous multiphoton excitation and, in particular, two-photon processes are now a standard technique in absorption spectroscopy and the concept of two-photon absorption cross section is frequently used to determine the two-photon transition probability [I?]. These cross sections are based on an extension of the “golden rule” obtained by second order time dependent perturbation methods. For example, for a two-photon process in which transitions between levels 1 and 3 through an intermediate state 2 are induced (see fig. l), the transition probability is given by [ 131 cross
With the advent of the laser, coherent light sources emitting ultrashort (ps) pulses have become available. Using these pulses one is able to study coherence phenomena such as self-induced transparency (SIT) [I], optical nutation effects [2] and photon echoes [3) _ The experimental conditions under which these phenomena can be observed require intense light signals molecular
PROCESSES
times
with
The relevant
the atomic time
scale
or the is determin-
time, T2, of the system and for most systems one has to use solid state mode-locked lasers in order to observe coherence phenomena on this time scale [4,5]. Coherent interaction in most cases means that the transition probability per atom cannot be described by a time independent quantity, such as is expressed by the Fermi “golden rule” and the Einstein B coefficient. The “golden rule” is obtained by using time-dependent perturbation methods which are limited to weak light signals and relatively long interaction times. For coherent interactions these conditions are not fulfilled. Rather, one should use a method similar to that used by Rabi to derive the “strong signal theory” [6], which generally yields periodic transition probabilities [7-g]. Intense lasers have extended the studies of conventional flash photolysis to shorter time scales [lo]. These involve multiphoton consecutive transitions, which are usually treated kinetically using absorption ed by the dephasing
WI3 =(nlSfi4)
I&H’,,
IZ P(W)/(W--o,#,
(1)
where H& s MlH’ I I ), H’-is the perturbaiion and p(o) the lirie shape Function. In the framework of the “golden rule” there is a clear distinction between simultaneous absorption of two photons and a consecutive one in which level 2 is also in resonance with the radiation field at frequency w. It is obvious that on resonance w12 = w and WI3 + 00. This difficulty is
‘: .’
: _.
:
‘_
. .
: .-..
:
._. :_
.:.
-.
J. Kotriel. S. Speiser/Coherer;t
292
spectrostipy 1171. Choosing the zero of energy at level 1, the hamiltonian For this system neglecting one photon coupling of levels 1 and 3 is
3hw
2
il ---
----
multiphoton absorption
H=f!f,+k&=
(fiw+fi)
a572 +2izwn~n3
+Rwctc
8 +E,2(=~a,C+nt12aci)+e2~(nT3azC+~~~JCi).
hw
where S measures
the deviation
(2)
from resonance
of
level 2, a:and Q,. are the creation and annihilation operators of thejth state, ct and c the corresponding
I-I
F&. 1. Schematic level diagxun describing mherenr two pho-
operators for photons, and enln is the interaction coupling coefficient for the levels )?I and II. The physically relevant space is spanned by the zero-order states
ton processes.
usually circumvented by introdu&ng phenomenological damping terms, which allow us to treat consecutive processes kinetically using separate transition probabilities for the transitions 1 + 2 and 2 + 3. Recently, an attempt was made [13] to unify both approaches treating simultaneous transitions by kinetic methods, defining the simultaneous process as being the limiting case of a consecutive transition for lifetimes 72 ‘+ Cl of the intermediate level. Moreover, using this treatment a zaturation law for two-photon processes was obtained. However, experimentally saturation is expected to be observed only at the very high light signals for which the “goklen rule” is not applicable. Just as for one-photon processes we should here consider two-photon coherence phenomena. This was done for example in studies of propagation effects such as two-photon SIT [14-161. However, transition probabilities fdr coherent multipho-, ton processes are not available. Such quantities are obtained in the following sections by an extension of Rabi’s “strong signal theory”.
xi = [(N-j+1)!]-‘/2
(@-it!
lj,O,,
j= 1,2,3,
(3)
where lj,O) indicates that the system is in the jthlevel and that the field is in the vacuum state. In order to obtain the stationary states of the complete hamiltonian we construct its matrix representation,
Iwo+6 E23
‘h=
The corresponding
eigenvalues
E,
E, = Miw.
=h’?w-
g_,
(4)
are E3 =hrJiw+t+,
(5)
with the eigenfunctions 41 =-(&1x, Q52 =
+.5-x2
(-azxl
+a2x3)/P-
I
+~,x3)lP, (6)
2. Coherent
two-photon
procgses
The interaction of a monochromatic field consisting of N photons_having. frequency w with a threelevel atomic system is studied. The interaction involves induced resonant two-photon transitions between the ground state level 1 and the final state level 3 through Come virtual state which involves temporal non-resonant coupl‘ing. determined by the uncertainty principle with-some level 2. The situation is depicted in fig: 1. A iel.ated system is discussed by Shimoda and Shimizu &I .their extensive review on non linear
p
= (a; +&1’2,
E, = (A F 6)/Z,
A = [a2 +4p2]1’2
P+ =
,
co2+ ty2
(7)
Initially. at f =O, the system is in the state x(0)=x1_ In order to follow the.time evolution df the system we express x1 in terms qf the stationary eigenstates
XI = -
[-,8+@‘1+&+2
(8)
+&+~rt8-@~]l(fl~A),
thus
:. : _.:..; .:
..
“..
:-
: ::
:,
,‘.
.,
..
_
-..
..._
J.
Katriel. S.
SpeisedCohcrent
multiphoron
P,(t) f Bat&
+a:&exp
(it+rfi)
t+.$- [exp (it+@)
+a:$+exp(-i.$--t/h)]
-exp(-i&+)1
+(Y~(Y*[E_exp(iE&)+,$+
exp(--i&r/fi)-
PI(r)=
{(a; +a;)
A] x3)
of the system are
[1-cos(A-
r/e)]
+(t)
= {I -(/3/A)2 -
[l -cos(A.
(11) f,%)]
[~-~s(~+~/~)+~+cos(~-t/~)l/Al 2bQlla2/P212.
The population probabilities vary with a periodicity depending on the relative magnitudes of the system parameters. This is of course different from the result obtained through the use of perturbation methods manifested by the “golden rule” result, eq. (1). Moreover, the transition 1 --z 2 has a fiite probability which increases as one approaches resonance, S = 0. The value of 6 can be used to determine the time scale in which coherence is maintained in the system, since it measures the uncertainty in energy for the excitation of a virtual level. The coherence time, T2 should be larger than the period of P*(t) which is given by -ii/A. These two uncertainties do fulfill the uncertainty principle At*AE
= co2
+ @2)“2 G fi.
The probability for the transition 1 + 3 is measured by P,(r), and we note that in contrast with the “‘golden rule” probability no difficulties arise for a resonant interaction through a real intermediate level, When 6 = 0, we obtain
(Pf/fi) =
cos2(q2&G/fi),
P*(f) = sin’ @f/h) = sin” (e12flr/fi),
P#) = 0,
(12’)
which are the well known e12 = (Po/&‘)
+Ba:cr?_/A)[f+cos(~-~/~)+~-cos(~+~/~)l)/p4, P,(t)=2(~+A)~[l-cos(A.r/fi)],
It is gratifying to note that if the third level is not coupled, i.e., ~73 =O, then eq. (12) becomes P,(r)
(10)
probabilities
-2(a:fi/A)2
X,
x2
X [exp(-iiwN~)/(/3ZA)]. Hence the population
+Ofllb*I* ,
(93
Using eq. (6) we obtain
-al
={[a;cm(bt/ft)
t/h)
exp(-iE2
+ <+qP-43 exp (- iE, @i)l(B2A).
x(r)={[crsA
293
absorprion
Rabi formulae
[7,8], with
D r2 = (8=h~/V)l’~D~~,
(13)
where E. is the maximum amplitude of the eIectric field, D,2 = e( I[ r12) is the transition dipole moment coupling the levels I and 2, and V is the interaction volume [IS]. Inspection of eq. (12) shows that the system is periodically in its ground state at times f =Zlrhz/P; (n = 0, I ,2,. . .). For cases in which a2 < aI, one finds that at times given by cos(@t/fi) = -- (u~/cY~)~ : P, =o,
Pz = (CYf- a;>/u;,
P3 = OY2/q2.
(14)
This means that the ground state is periodically completely depopulated. However, complete transient transparency is reached in the following situations: (1)crI =or2whichallowsforP1=P2=O;P3=1 for pf//t = (2n + 1) x, this is depicted in fg. 2. (2) ot /a2 = fi It 1 which allows for P, =Pz = P3 = l/3 for sin2(@/fi)= (4 f a/6. Thus, two-photon SIT is possible in many situations. The two-photon process can be a consecutive or a simultaneous event, without a clear distinction between the two for coherent excitations. The transition from a consecutive process to a simultaneous absorption of two-photons is very smooth. It is obvious that for an extreme off-resonant coupling of level 2, i.e., forS+=,weobtainP, =l; P2=P3=Owhichisa result obtained also from eq. (1). However, we do not have to treat cases where 6 =O separately since they represent a special case of a more general formulation.
.. :
‘.
.:.
.. ..
: ,
.,
_‘.
J. Kitrid,
S. SpeisqlCoherolt
multiphoton
absorption
”
I. Fig. 2. Temporal
behwiour
Fig. 3. Schematic level diagram describing resonant coherent multiphoton processes.
Nyl/h)?
of transition probabilities
Car rc-
sonant two-photon induced coherent transitions in a three level system for equal lcvel coupling terms.
3. Multiphoton
resonmt
where x = (fiw - E)/E&. This is the Hueckel for a linear polyene [ 191. The eigenvalues are
coherent processes
matrix
Xj=2COS(jn/n+l),
Having shown that resonant and off-resonant coherent two-photon proLesses have similar features, we shall discuss multiphoton coherent processes for a resonant interaction between a radiation field and a multilevel system, as shown in fig. 3. The non-vnulshing matrix elements of the hamiltonian for this system are Hii =Aw,
(j=
Jfi,j+,=Hi+,j=~l:i+l
Ei =ftw
-
“j = fi
(15)
We shali solve the eigenvalue problem for the case of IV> j and equal coupling for all levels, i.e., eLj+l = E ‘for all i’s. IIIUS, Hi.i+i = Efi.and the secular equation is given by
(17)
ti’12xj_
The eigenstates
are given by
@rsjr 3
=
sjr= [2/h
1,2 ,..._ 11) [IV-(j-l)!]‘/‘.
i.e.,
“’ sin [jnr/(n + 1)] . + 111
Follotiingsimilar steps to those leading to eq. (12) we find that the time evolution of the system initially at state s is given by @JO = exp(-i4
$t
&r( 2
Sj&
erp Wr)),
09)
where e = (247fi)
cos [jnl(n
Thus the probability r is
+ l)] -
of finding the system
in any state
J. Katrici. S. SpeirerfCoRcrent
where
multiphotor~
absorption
29.5
References
7 r-F.5= 0,
= n/2,
(v + s) even, .(r + S) odd.
Usually, the system is initially at level s= I and we are interested in P,,(t). However. eq. (20) can also describe the probability with which a system. initially prepared in an excited state (as in a chemical laser system). makes coherent transitions to all other levels.
4. Conrlusions It has been shown that transition probabilities for multiphoton coherent processes differ from the widely used “‘golden rule” probabilities. For laser pulses longer than T2 and for relatively low fields one can still use the concept of a cross section derived by timedependent perturbation methods. However, for extremely short, intense pulses, such as those produced by modelocked lasers, there will be systems for which the present treatment should be used. It predicts SIT for a multiphoton process in certain conditions and also the possibility of exciting the upper state of a multistate system coherently and effectively through a consecutive process. This may serve as an explanation of the observed photochemistry in systems excited by infrared lasers, a process requiring an efficient multiphoton absorption of an anharmonic oscillator per dissociated molecule [20].
[ 11 S.L. McCall and E.L. Ilahn, Phys. Rev. 183 (1969) 457. [2] G.B. Hacker and CL. Tang, Phys. Rev. 184 (1969) 356. [3] N.A. Kumit, I.D. Abcllaand S.R. Homan. Phys. Rev. 141 (1966) 391. [4] C.L. Lamb, Rev. Mod. Phys. 43 (1971) 99. [5] E. Cour~cns.in: Laser Handbook. eds. E.0. SchultzDubois and F.T. Arccohi (Nortll-Holland. Amsterdam, 1972) pp. 1259-1322. 161 1.1. Rabi. Phys. Rev. 51 (1937) 652. 17 1 A. Ynriv. Quantum Electronics (Wiley, New York. 1967) pp. 213-217. [S] A. Blaitland and M.H. Dunn, Laser Physics (NorthIlolland. Amsterdam. 1969) pp. 80-84. 191 W.R. Salzman. Phys. Rev. A4 (1971) 1530; Phys. Rev. A5 (1972) 789. [IO] G. Porter and M.A. West. Techniques of Chemistry. VI (II), Chap. X Wiley-Interscience, N.Y.. 1974). ]I 1] S. Speiscr, R. van der Werf and J. Kommnndcur, Chcm. Phys. 1 (1973) 297. [ 121 J.M. Warlock, in: Laser Handbuok. rds. E.O. SchultzDubois and F.T. Arecchi (North-Ifalland, Amsterdam 1972) pp. 1323-1369. 1131 0. Kaliiand S. Kimel.Chcm. Phys.5’(1974)488. 114) M. Takatsuji. Phys. Rev. All (1975) 634. [ 15 1 N. Tan-no, J. Yokoto and H. Inaba. Phys. Rev. Letters 29 (1972) 1211. [16] E. Hannmura. J. Phys. Sot. Jnp. 6 (1974) 1596. 1171 K. Shimoda and T. Shimizu. Prog. Quant. Electron 2 (1972)45 [IS] Set ref. [7] p. 207. 1191 E-z-, F.L. Pilar. Elementary (McGraw-Hill, New York. 1201 P.L. Robinson.
Quantum Chemistry 1968) pp. 593-595.
Laser isotope separation, to bc published in Proceedings of the Second Laser Spectroscopy Conference, hlcgdvc (T-‘rzmce, 1975).
Publishing
TRANSlTtON
Company
PROBABILITIES
R’eccived 4 September
FOR COHERENT
MULTIPHOTON
ABSORPTION
1975
The feusibility of coherent multiphoron propn_=tion effects soch as two-photon self-induced by calcularinp the coherent trnnsirion probabilities for a multiphoton process. For a two-photon system, periodic probability functions are obtained which increusc smoothly as the intermediate with the radiation field. The results for a coherent multiphoton excitation of a multilevel system “strong signal theory” Tar a one-photon excikrion of 3 two-level system.
1. Introduction
and short
interaction system.
transparency is examined excitation of a three-level state approaches resonance arc an extension of RPbi’s
sections based on the “golden rule” [I I] It is obvious that there are experimental conditions in which multiphoton consecutive colzcrer~~ transitions should be taken into account. Moreover, the coherence time (the dephasing time T2) of a complex system may be very short and even comparable to the uncertainty in time during the interaction with the radiation field. Thus, as will be shown beIow, no cIear distinction can be made in such cases between consecutive and simultaneous multiphoton transitions. Simultaneous multiphoton excitation and, in particular, two-photon processes are now a standard technique in absorption spectroscopy and the concept of two-photon absorption cross section is frequently used to determine the two-photon transition probability [I?]. These cross sections are based on an extension of the “golden rule” obtained by second order time dependent perturbation methods. For example, for a two-photon process in which transitions between levels 1 and 3 through an intermediate state 2 are induced (see fig. l), the transition probability is given by [ 131 cross
With the advent of the laser, coherent light sources emitting ultrashort (ps) pulses have become available. Using these pulses one is able to study coherence phenomena such as self-induced transparency (SIT) [I], optical nutation effects [2] and photon echoes [3) _ The experimental conditions under which these phenomena can be observed require intense light signals molecular
PROCESSES
times
with
The relevant
the atomic time
scale
or the is determin-
time, T2, of the system and for most systems one has to use solid state mode-locked lasers in order to observe coherence phenomena on this time scale [4,5]. Coherent interaction in most cases means that the transition probability per atom cannot be described by a time independent quantity, such as is expressed by the Fermi “golden rule” and the Einstein B coefficient. The “golden rule” is obtained by using time-dependent perturbation methods which are limited to weak light signals and relatively long interaction times. For coherent interactions these conditions are not fulfilled. Rather, one should use a method similar to that used by Rabi to derive the “strong signal theory” [6], which generally yields periodic transition probabilities [7-g]. Intense lasers have extended the studies of conventional flash photolysis to shorter time scales [lo]. These involve multiphoton consecutive transitions, which are usually treated kinetically using absorption ed by the dephasing
WI3 =(nlSfi4)
I&H’,,
IZ P(W)/(W--o,#,
(1)
where H& s MlH’ I I ), H’-is the perturbaiion and p(o) the lirie shape Function. In the framework of the “golden rule” there is a clear distinction between simultaneous absorption of two photons and a consecutive one in which level 2 is also in resonance with the radiation field at frequency w. It is obvious that on resonance w12 = w and WI3 + 00. This difficulty is
‘: .’
: _.
:
‘_
. .
: .-..
:
._. :_
.:.
-.
J. Kotriel. S. Speiser/Coherer;t
292
spectrostipy 1171. Choosing the zero of energy at level 1, the hamiltonian For this system neglecting one photon coupling of levels 1 and 3 is
3hw
2
il ---
----
multiphoton absorption
H=f!f,+k&=
(fiw+fi)
a572 +2izwn~n3
+Rwctc
8 +E,2(=~a,C+nt12aci)+e2~(nT3azC+~~~JCi).
hw
where S measures
the deviation
(2)
from resonance
of
level 2, a:and Q,. are the creation and annihilation operators of thejth state, ct and c the corresponding
I-I
F&. 1. Schematic level diagxun describing mherenr two pho-
operators for photons, and enln is the interaction coupling coefficient for the levels )?I and II. The physically relevant space is spanned by the zero-order states
ton processes.
usually circumvented by introdu&ng phenomenological damping terms, which allow us to treat consecutive processes kinetically using separate transition probabilities for the transitions 1 + 2 and 2 + 3. Recently, an attempt was made [13] to unify both approaches treating simultaneous transitions by kinetic methods, defining the simultaneous process as being the limiting case of a consecutive transition for lifetimes 72 ‘+ Cl of the intermediate level. Moreover, using this treatment a zaturation law for two-photon processes was obtained. However, experimentally saturation is expected to be observed only at the very high light signals for which the “goklen rule” is not applicable. Just as for one-photon processes we should here consider two-photon coherence phenomena. This was done for example in studies of propagation effects such as two-photon SIT [14-161. However, transition probabilities fdr coherent multipho-, ton processes are not available. Such quantities are obtained in the following sections by an extension of Rabi’s “strong signal theory”.
xi = [(N-j+1)!]-‘/2
(@-it!
lj,O,,
j= 1,2,3,
(3)
where lj,O) indicates that the system is in the jthlevel and that the field is in the vacuum state. In order to obtain the stationary states of the complete hamiltonian we construct its matrix representation,
Iwo+6 E23
‘h=
The corresponding
eigenvalues
E,
E, = Miw.
=h’?w-
g_,
(4)
are E3 =hrJiw+t+,
(5)
with the eigenfunctions 41 =-(&1x, Q52 =
+.5-x2
(-azxl
+a2x3)/P-
I
+~,x3)lP, (6)
2. Coherent
two-photon
procgses
The interaction of a monochromatic field consisting of N photons_having. frequency w with a threelevel atomic system is studied. The interaction involves induced resonant two-photon transitions between the ground state level 1 and the final state level 3 through Come virtual state which involves temporal non-resonant coupl‘ing. determined by the uncertainty principle with-some level 2. The situation is depicted in fig: 1. A iel.ated system is discussed by Shimoda and Shimizu &I .their extensive review on non linear
p
= (a; +&1’2,
E, = (A F 6)/Z,
A = [a2 +4p2]1’2
P+ =
,
co2+ ty2
(7)
Initially. at f =O, the system is in the state x(0)=x1_ In order to follow the.time evolution df the system we express x1 in terms qf the stationary eigenstates
XI = -
[-,8+@‘1+&+2
(8)
+&+~rt8-@~]l(fl~A),
thus
:. : _.:..; .:
..
“..
:-
: ::
:,
,‘.
.,
..
_
-..
..._
J.
Katriel. S.
SpeisedCohcrent
multiphoron
P,(t) f Bat&
+a:&exp
(it+rfi)
t+.$- [exp (it+@)
+a:$+exp(-i.$--t/h)]
-exp(-i&+)1
+(Y~(Y*[E_exp(iE&)+,$+
exp(--i&r/fi)-
PI(r)=
{(a; +a;)
A] x3)
of the system are
[1-cos(A-
r/e)]
+(t)
= {I -(/3/A)2 -
[l -cos(A.
(11) f,%)]
[~-~s(~+~/~)+~+cos(~-t/~)l/Al 2bQlla2/P212.
The population probabilities vary with a periodicity depending on the relative magnitudes of the system parameters. This is of course different from the result obtained through the use of perturbation methods manifested by the “golden rule” result, eq. (1). Moreover, the transition 1 --z 2 has a fiite probability which increases as one approaches resonance, S = 0. The value of 6 can be used to determine the time scale in which coherence is maintained in the system, since it measures the uncertainty in energy for the excitation of a virtual level. The coherence time, T2 should be larger than the period of P*(t) which is given by -ii/A. These two uncertainties do fulfill the uncertainty principle At*AE
= co2
+ @2)“2 G fi.
The probability for the transition 1 + 3 is measured by P,(r), and we note that in contrast with the “‘golden rule” probability no difficulties arise for a resonant interaction through a real intermediate level, When 6 = 0, we obtain
(Pf/fi) =
cos2(q2&G/fi),
P*(f) = sin’ @f/h) = sin” (e12flr/fi),
P#) = 0,
(12’)
which are the well known e12 = (Po/&‘)
+Ba:cr?_/A)[f+cos(~-~/~)+~-cos(~+~/~)l)/p4, P,(t)=2(~+A)~[l-cos(A.r/fi)],
It is gratifying to note that if the third level is not coupled, i.e., ~73 =O, then eq. (12) becomes P,(r)
(10)
probabilities
-2(a:fi/A)2
X,
x2
X [exp(-iiwN~)/(/3ZA)]. Hence the population
+Ofllb*I* ,
(93
Using eq. (6) we obtain
-al
={[a;cm(bt/ft)
t/h)
exp(-iE2
+ <+qP-43 exp (- iE, @i)l(B2A).
x(r)={[crsA
293
absorprion
Rabi formulae
[7,8], with
D r2 = (8=h~/V)l’~D~~,
(13)
where E. is the maximum amplitude of the eIectric field, D,2 = e( I[ r12) is the transition dipole moment coupling the levels I and 2, and V is the interaction volume [IS]. Inspection of eq. (12) shows that the system is periodically in its ground state at times f =Zlrhz/P; (n = 0, I ,2,. . .). For cases in which a2 < aI, one finds that at times given by cos(@t/fi) = -- (u~/cY~)~ : P, =o,
Pz = (CYf- a;>/u;,
P3 = OY2/q2.
(14)
This means that the ground state is periodically completely depopulated. However, complete transient transparency is reached in the following situations: (1)crI =or2whichallowsforP1=P2=O;P3=1 for pf//t = (2n + 1) x, this is depicted in fg. 2. (2) ot /a2 = fi It 1 which allows for P, =Pz = P3 = l/3 for sin2(@/fi)= (4 f a/6. Thus, two-photon SIT is possible in many situations. The two-photon process can be a consecutive or a simultaneous event, without a clear distinction between the two for coherent excitations. The transition from a consecutive process to a simultaneous absorption of two-photons is very smooth. It is obvious that for an extreme off-resonant coupling of level 2, i.e., forS+=,weobtainP, =l; P2=P3=Owhichisa result obtained also from eq. (1). However, we do not have to treat cases where 6 =O separately since they represent a special case of a more general formulation.
.. :
‘.
.:.
.. ..
: ,
.,
_‘.
J. Kitrid,
S. SpeisqlCoherolt
multiphoton
absorption
”
I. Fig. 2. Temporal
behwiour
Fig. 3. Schematic level diagram describing resonant coherent multiphoton processes.
Nyl/h)?
of transition probabilities
Car rc-
sonant two-photon induced coherent transitions in a three level system for equal lcvel coupling terms.
3. Multiphoton
resonmt
where x = (fiw - E)/E&. This is the Hueckel for a linear polyene [ 191. The eigenvalues are
coherent processes
matrix
Xj=2COS(jn/n+l),
Having shown that resonant and off-resonant coherent two-photon proLesses have similar features, we shall discuss multiphoton coherent processes for a resonant interaction between a radiation field and a multilevel system, as shown in fig. 3. The non-vnulshing matrix elements of the hamiltonian for this system are Hii =Aw,
(j=
Jfi,j+,=Hi+,j=~l:i+l
Ei =ftw
-
“j = fi
(15)
We shali solve the eigenvalue problem for the case of IV> j and equal coupling for all levels, i.e., eLj+l = E ‘for all i’s. IIIUS, Hi.i+i = Efi.and the secular equation is given by
(17)
ti’12xj_
The eigenstates
are given by
@rsjr 3
=
sjr= [2/h
1,2 ,..._ 11) [IV-(j-l)!]‘/‘.
i.e.,
“’ sin [jnr/(n + 1)] . + 111
Follotiingsimilar steps to those leading to eq. (12) we find that the time evolution of the system initially at state s is given by @JO = exp(-i4
$t
&r( 2
Sj&
erp Wr)),
09)
where e = (247fi)
cos [jnl(n
Thus the probability r is
+ l)] -
of finding the system
in any state
J. Katrici. S. SpeirerfCoRcrent
where
multiphotor~
absorption
29.5
References
7 r-F.5= 0,
= n/2,
(v + s) even, .(r + S) odd.
Usually, the system is initially at level s= I and we are interested in P,,(t). However. eq. (20) can also describe the probability with which a system. initially prepared in an excited state (as in a chemical laser system). makes coherent transitions to all other levels.
4. Conrlusions It has been shown that transition probabilities for multiphoton coherent processes differ from the widely used “‘golden rule” probabilities. For laser pulses longer than T2 and for relatively low fields one can still use the concept of a cross section derived by timedependent perturbation methods. However, for extremely short, intense pulses, such as those produced by modelocked lasers, there will be systems for which the present treatment should be used. It predicts SIT for a multiphoton process in certain conditions and also the possibility of exciting the upper state of a multistate system coherently and effectively through a consecutive process. This may serve as an explanation of the observed photochemistry in systems excited by infrared lasers, a process requiring an efficient multiphoton absorption of an anharmonic oscillator per dissociated molecule [20].
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Laser isotope separation, to bc published in Proceedings of the Second Laser Spectroscopy Conference, hlcgdvc (T-‘rzmce, 1975).