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Multiphoton absorption by atoms
Volume 48A, number 5
PHYSICS LETTERS
15 July 1974
MULTIPHOTON ABSORPTION BY ATOMS B.J. CHOUDHURY Department of Physics. University of Manitoba, Winnipeg, Canada Received S May 1974 A theory of multiphoton transition in atoms by a reasonably strong electromagnetic field has been outlined. The theory treats the perturbation consistently and takes into account of the higher order processes associated with the multiphoton transitions approximately through the mean frequency approximation.
In the past, several authors have used the MTA [1] to study the intensity dependence of multiphoton processes. Recently, however, it has been pointed out by Cohen Tannoudji [2] and Decoster [3] that the MTA should not be used to calculate multiphoton transition amplitudes because the perturbation has not been treated properly. Decoster has pointed out that the MTA can be used to calculate a one photon transition amplitude by a dressed atom. The purpose of the present note is to give expression for the transition amplitude which can be used to calculate multiphoton transition in atoms. We will follow the procedure outlined by Choudhury [4] except for the definition of the s-matrix which has been stated incorrectly in that paper. The s-matrix element for the e.m. transition from a state to the state ~ is (II = c = 1) (S—1)f1= —i
f dt(Ø~.,H’exp (ieA~r)
where H’ =
(elm) A p; A
=
—
i
where H1
“~),
—
f exp {—iH0(t
a cos wt and —
(1)
is the solution of the
i~1
t’)}H1 ~LI~(t’)dt’,
integral
equation (2)
e(aA/at)r.
The calculation of the s-matrix element for the multiphoton process will be performed on the basis set of unperturbed energy states of the atom. Within this set the kth order s-matrix element is given by 2 [Enk_Ef] dt(Øf~exp(ieA~r)Øn~) J’(~nk,hhI1~1)thk, (3) (S—l)~(—i) where P 1~1is the (k—I )th order perturbation solution of eq. (2) and it has been defined in ref. [4]. In writing eq. (3) we have made use of the commutation relation
E
I
[H0,exp (ieA-r)] = —H’ exp (ieA -r), (4) 2/2m + V(r). where H0 p An analytic expression for P~~~calculated by the mean frequency approximation [5] has been given in [4] where the advantage and disadvantage of this approximation have also been discussed. Using this expression for P~’~in eq. (3) we obtain the transition amplitude for the N-photon absorption process as: T~N)= k=O
(5)
333
Volume 48A, number 5
PHYSICS LETTERS
15 July 1974
where Q~=—Nw(~f,JN(ea~r)~~)and Q(k~O)=~ k1D[~l
+ (1
—
—
~+(k_2n)W}~f’
(ea.rw/2)kJN+k 2n (ea-r)~~)
(~,(ea.rw/2)kJN+k
~÷+(~~fl~)
2n 2 (ea.r)Øi)].
J1(x) is the Bessel function of order 1 and ~2is the mean frequency of the atom as defined in [5]. The coefficients kDn satisfy the following recurrence relations fork> 2m but m~O: 1
+klDl
Lfl~(~+jw)
I
mj(kw_2mw+cl)
2m~ =~[2m~1Dm+2m_1Dmi],
kD=1/fl(~+fw) j1
kDm =kDk_m((~~_w)fork>2m.
It can be verified that for a low intensity e.m. field, the sum of first N-terms in eq. (5) is identical to the result of Bebb and Gold [5]. The following points need to be noted (a) Because eq. (5) has been obtained by a perturbation expansion of eq. (2), it is not valid for an arbitrarily intense e.m. field, (b) As is, eq. (5) should not be used when there is a resonance intermediate state and also the results calculated from eq. (5) will be approximate because of the mean frequency approximation. With considerably less numerical computation, it can be used to obtain a reasonable estimate of the transition probability and its intensity dependence beyond the linear region [6]. (In eq. (5) there is an implied restriction on the sum over the index k. The origin of this restriction can be traced from the matrix elements appearing in (3), where it is to be noted that the lowest order term ofthe matrix element (~,exp (ieA-r)q.fl~)which is non-zero corresponds to (Ø~,(ieA-r)4flk). An expression for the T-matrix element which incorporates this restriction can also be written in the form analogous to eq. (5). This result and its relation to the higher order perturbation theory result will be published). The author is indebted to Prof. H.R. Reiss for pointing out the error in ref. [4] and also for numerous discussions.
References [1] H.R. Reiss, Phys. Rev. D4 (1971) 3533. (2] C. Cohen Tannoudji et al., Phys. Rev. A8 (1973) 2747. [3] A. Decoster, Phys. Rev. A, to be published. [41 B.J. Choudhury, Phys. Rev. A8 (1973) 1759. [51 H.B. Bebb and A. Gold, Phys. Rev. 143 (1966) 1. (6] Y. Gontier and M. Trahin, Phys. Rev. A7 (1973) 1899.
334
PHYSICS LETTERS
15 July 1974
MULTIPHOTON ABSORPTION BY ATOMS B.J. CHOUDHURY Department of Physics. University of Manitoba, Winnipeg, Canada Received S May 1974 A theory of multiphoton transition in atoms by a reasonably strong electromagnetic field has been outlined. The theory treats the perturbation consistently and takes into account of the higher order processes associated with the multiphoton transitions approximately through the mean frequency approximation.
In the past, several authors have used the MTA [1] to study the intensity dependence of multiphoton processes. Recently, however, it has been pointed out by Cohen Tannoudji [2] and Decoster [3] that the MTA should not be used to calculate multiphoton transition amplitudes because the perturbation has not been treated properly. Decoster has pointed out that the MTA can be used to calculate a one photon transition amplitude by a dressed atom. The purpose of the present note is to give expression for the transition amplitude which can be used to calculate multiphoton transition in atoms. We will follow the procedure outlined by Choudhury [4] except for the definition of the s-matrix which has been stated incorrectly in that paper. The s-matrix element for the e.m. transition from a state to the state ~ is (II = c = 1) (S—1)f1= —i
f dt(Ø~.,H’exp (ieA~r)
where H’ =
(elm) A p; A
=
—
i
where H1
“~),
—
f exp {—iH0(t
a cos wt and —
(1)
is the solution of the
i~1
t’)}H1 ~LI~(t’)dt’,
integral
equation (2)
e(aA/at)r.
The calculation of the s-matrix element for the multiphoton process will be performed on the basis set of unperturbed energy states of the atom. Within this set the kth order s-matrix element is given by 2 [Enk_Ef] dt(Øf~exp(ieA~r)Øn~) J’(~nk,hhI1~1)thk, (3) (S—l)~(—i) where P 1~1is the (k—I )th order perturbation solution of eq. (2) and it has been defined in ref. [4]. In writing eq. (3) we have made use of the commutation relation
E
I
[H0,exp (ieA-r)] = —H’ exp (ieA -r), (4) 2/2m + V(r). where H0 p An analytic expression for P~~~calculated by the mean frequency approximation [5] has been given in [4] where the advantage and disadvantage of this approximation have also been discussed. Using this expression for P~’~in eq. (3) we obtain the transition amplitude for the N-photon absorption process as: T~N)= k=O
(5)
333
Volume 48A, number 5
PHYSICS LETTERS
15 July 1974
where Q~=—Nw(~f,JN(ea~r)~~)and Q(k~O)=~ k1D[~l
+ (1
—
—
~+(k_2n)W}~f’
(ea.rw/2)kJN+k 2n (ea-r)~~)
(~,(ea.rw/2)kJN+k
~÷+(~~fl~)
2n 2 (ea.r)Øi)].
J1(x) is the Bessel function of order 1 and ~2is the mean frequency of the atom as defined in [5]. The coefficients kDn satisfy the following recurrence relations fork> 2m but m~O: 1
+klDl
Lfl~(~+jw)
I
mj(kw_2mw+cl)
2m~ =~[2m~1Dm+2m_1Dmi],
kD=1/fl(~+fw) j1
kDm =kDk_m((~~_w)fork>2m.
It can be verified that for a low intensity e.m. field, the sum of first N-terms in eq. (5) is identical to the result of Bebb and Gold [5]. The following points need to be noted (a) Because eq. (5) has been obtained by a perturbation expansion of eq. (2), it is not valid for an arbitrarily intense e.m. field, (b) As is, eq. (5) should not be used when there is a resonance intermediate state and also the results calculated from eq. (5) will be approximate because of the mean frequency approximation. With considerably less numerical computation, it can be used to obtain a reasonable estimate of the transition probability and its intensity dependence beyond the linear region [6]. (In eq. (5) there is an implied restriction on the sum over the index k. The origin of this restriction can be traced from the matrix elements appearing in (3), where it is to be noted that the lowest order term ofthe matrix element (~,exp (ieA-r)q.fl~)which is non-zero corresponds to (Ø~,(ieA-r)4flk). An expression for the T-matrix element which incorporates this restriction can also be written in the form analogous to eq. (5). This result and its relation to the higher order perturbation theory result will be published). The author is indebted to Prof. H.R. Reiss for pointing out the error in ref. [4] and also for numerous discussions.
References [1] H.R. Reiss, Phys. Rev. D4 (1971) 3533. (2] C. Cohen Tannoudji et al., Phys. Rev. A8 (1973) 2747. [3] A. Decoster, Phys. Rev. A, to be published. [41 B.J. Choudhury, Phys. Rev. A8 (1973) 1759. [51 H.B. Bebb and A. Gold, Phys. Rev. 143 (1966) 1. (6] Y. Gontier and M. Trahin, Phys. Rev. A7 (1973) 1899.
334