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Raman process in the case of multiphoton excitation of an helium atom in an intense laser beam
Volume 57A, number I
PHYSICS LETTERS
17 May 1976
RAMAN PROCESS IN THE CASE OF MULTIPHOTON EXCITATION OF AN HELIUM ATOM IN AN INTENSE LASER BEAM U. DEVI and M. MOHAN Department of Physics and Astrophysics, University of Delhi, Delhi-I/0007, India Received 10 September 1975 The transition probability of the Ranian process in the case of niultiphoton excitation of an He atom has been calculated using the space translation method.
The difficulty with perturbation theory calculations done earlier [11 is that it is applied to intensities far beyond 2/sec) of the usual theory and also the effect of the intense em. the limit of validity (photon flux ~ 1O~ph./cm wave on the initial state has been totally negelcted. In the present paper the reduced transition probability of multiphoton excitation of an He atom under the influence of an intense laser beam with the emission of one Raman photon of frequency w’ has been calculated using space translation method (in order to conserve the energy in the process Ef E~= Nw w’, where w is the frequency of photons belonging to intense field A, N is the number of photons of frequency w absorbed in the process and w’ is the frequency of the Raman photon corresponding to weak field A’). Natural units 11 = c = I and gauge A 0 = 0 and V A = 0 will be used here. The space translation method consists of a transformation to an accelerated frame as in ref. t2] The Schrodinger equation for a two electron systeni He atom in presence of em. field A is iaml.i/at=H~(r1r2t)
(I)
where the Hamiltonian is
2+ V. 1p1
H=
—
eA(t)
eA’(t)I
[p2
2 +
—
eA(t)
-~
eA’(t)~
Subscripts I and 2 describe the two electrons of the He atom. The interaction Hamiltonian is
H 1(t) =
-~-
[A(t) + A’(t)I ‘pi
-
[A(t) +A’(t)I
The ~12 term contributes negligible in the present process. We apply space translation operation (2)
~i(r1r2t) U~(r1r2t) where we take
~
~=
f f(A(t’)
+
A’(t’))• p1
+
(A(t’) + A’(t’)) P2} dt’l
(j)
when both the electrons are being excited simultaneously. When one electron at a time is being excited then
u
=
exp
[~f(A(t’)
+ A’(t’))
p1 dt’~
or
U= exp [ie
(ii)
f (A(t’)
+
A’(t’)) p~dt’]
Volume
57A, number 1
PHYSICS LETTERS
Non relativistic dipole approximation condition (w/c)a0 Define nE sin wt
t
e =
p
,
,
A(t ) dt
—
mJ
ea~ sin wt, mw .
=
—
‘~
,~,.
,
c~r sin w t
—
1 is used and A
=
17 May 1976
=
~ea cos wt and A’
=
~‘
ea’ cos w’t.
ea~ sin w i-. mw
—
—,-
The transition amplitude Afi between the unperturbed states ctm~ and ~ is defined [3] as: 1 A~=2iri ~ö(Ef—E~+Nw— w’)T~’ where T~’1is an term in (ØfH 1 m1i~>proportional to exp(iNwt) exp(—iw’t). Let M denote the matrix element 5fU~mi>.
M = (~fH1~> = (ØfH1 U~>= Using the relation exp(~sin wt) = ~
~
—
(3)
~i) (q
J~(z)exp(inwt)
in eq. (3) for both the cases (i) and (ii) we get respectively ~
M=(Ef—E~)
,,
<~f(P 1P2)IJ0(
~
P1)J~2(
p2)J~(csE p1)J~’(n’~’p2)Iø~(p1p2)>
~
X exp(iNwt) exp(—iw’t)
(A)
and M = (Ef
E1) ~
—
~1(p1 p~)IJ01(ia- p1)J~.(cs’~’p~)IØ~(p~ p2)> exp(iNwt) exp(---iw’t)
ni,n
(Ef
+
—
~
E1)
~Øt(Pi p2)IJ~,(~p2)J0(~’~’P2)Iø~(Pip2)> exp(iNwt) exp(—iw’t).
(B)
nI’n
If ç~m~and ç~jare symmetrical w.r.t. p1 and p2 then eq. (B) becomes M = 2(Ef
—
~
E1)
~Øf(Pl p2)1J01 (ci ~ p1)J~’(a’~’p~)IØ~(p~ P2)) exp(i~Vwt)exp(—iw’t).
(4)
n
Now since ma’ belongs to veiy weak field we retain only the J0 and J1 functions of a’~’ p1 and cs’~’ p2. We knowJ0(a’~’ p) 1, and J1(a’~’ p) ~-(ci’~’ p). Now by hypothesis the ~ term cannot satisfy energy conservation requirements and it will therefore not contribute. Therefore eq. (A) gives 1~ (E._E.) ~ (Øf(p T~’ 1P2)IJfl(aEp1)Jfl(maEp2)aE p1a’~’ p21c51(p1p2)) (5) and eq. (B) gives 1 (E~ E~) —
T~’
(~f(P1p
2)IJ~,(cia p1)a’~’ p1 I ~i(Pl P2)>.
(6)
In the limit of low intensity eq. (6) becomes
39
Volume 57A, number 1
T~’t= (Ef-- E~)~~t(PiP2)1 ~a
PHYSICS LETTERS
17 May 1976
p 1 a’~’
~~(p1P2)>~
(7)
The method is applicable to multiphoton processes involving arbitrary intensities and frequencies of the incidence radiation under the condition (w/c)a0 ~ 1. We are thankful to Professor SN. Biswas for many valuable discussions and encouragement.
References III
LV. Keldysh, Zh. Eksp. I. Teor. Fiz. 47 (1964) 1945; [Soy. Phys. JETP 20 (1965) 13071; A. Gold and H.B. Bebb, Phys. Rev. Letters 14 (1965) 60. [21 W.C. Renneberger, Phys. Rev. Letters 21(1968) 838. 131 F.I-I.M. laisal, J. Phys. B., Atom Molec. Phys. 6 (1973) L89.
40
PHYSICS LETTERS
17 May 1976
RAMAN PROCESS IN THE CASE OF MULTIPHOTON EXCITATION OF AN HELIUM ATOM IN AN INTENSE LASER BEAM U. DEVI and M. MOHAN Department of Physics and Astrophysics, University of Delhi, Delhi-I/0007, India Received 10 September 1975 The transition probability of the Ranian process in the case of niultiphoton excitation of an He atom has been calculated using the space translation method.
The difficulty with perturbation theory calculations done earlier [11 is that it is applied to intensities far beyond 2/sec) of the usual theory and also the effect of the intense em. the limit of validity (photon flux ~ 1O~ph./cm wave on the initial state has been totally negelcted. In the present paper the reduced transition probability of multiphoton excitation of an He atom under the influence of an intense laser beam with the emission of one Raman photon of frequency w’ has been calculated using space translation method (in order to conserve the energy in the process Ef E~= Nw w’, where w is the frequency of photons belonging to intense field A, N is the number of photons of frequency w absorbed in the process and w’ is the frequency of the Raman photon corresponding to weak field A’). Natural units 11 = c = I and gauge A 0 = 0 and V A = 0 will be used here. The space translation method consists of a transformation to an accelerated frame as in ref. t2] The Schrodinger equation for a two electron systeni He atom in presence of em. field A is iaml.i/at=H~(r1r2t)
(I)
where the Hamiltonian is
2+ V. 1p1
H=
—
eA(t)
eA’(t)I
[p2
2 +
—
eA(t)
-~
eA’(t)~
Subscripts I and 2 describe the two electrons of the He atom. The interaction Hamiltonian is
H 1(t) =
-~-
[A(t) + A’(t)I ‘pi
-
[A(t) +A’(t)I
The ~12 term contributes negligible in the present process. We apply space translation operation (2)
~i(r1r2t) U~(r1r2t) where we take
~
~=
f f(A(t’)
+
A’(t’))• p1
+
(A(t’) + A’(t’)) P2} dt’l
(j)
when both the electrons are being excited simultaneously. When one electron at a time is being excited then
u
=
exp
[~f(A(t’)
+ A’(t’))
p1 dt’~
or
U= exp [ie
(ii)
f (A(t’)
+
A’(t’)) p~dt’]
Volume
57A, number 1
PHYSICS LETTERS
Non relativistic dipole approximation condition (w/c)a0 Define nE sin wt
t
e =
p
,
,
A(t ) dt
—
mJ
ea~ sin wt, mw .
=
—
‘~
,~,.
,
c~r sin w t
—
1 is used and A
=
17 May 1976
=
~ea cos wt and A’
=
~‘
ea’ cos w’t.
ea~ sin w i-. mw
—
—,-
The transition amplitude Afi between the unperturbed states ctm~ and ~ is defined [3] as: 1 A~=2iri ~ö(Ef—E~+Nw— w’)T~’ where T~’1is an term in (ØfH 1 m1i~>proportional to exp(iNwt) exp(—iw’t). Let M denote the matrix element 5fU~mi>.
M = (~fH1~> = (ØfH1 U~>= Using the relation exp(~sin wt) = ~
~
—
(3)
~i) (q
J~(z)exp(inwt)
in eq. (3) for both the cases (i) and (ii) we get respectively ~
M=(Ef—E~)
,,
<~f(P 1P2)IJ0(
~
P1)J~2(
p2)J~(csE p1)J~’(n’~’p2)Iø~(p1p2)>
~
X exp(iNwt) exp(—iw’t)
(A)
and M = (Ef
E1) ~
—
~1(p1 p~)IJ01(ia- p1)J~.(cs’~’p~)IØ~(p~ p2)> exp(iNwt) exp(---iw’t)
ni,n
(Ef
+
—
~
E1)
~Øt(Pi p2)IJ~,(~p2)J0(~’~’P2)Iø~(Pip2)> exp(iNwt) exp(—iw’t).
(B)
nI’n
If ç~m~and ç~jare symmetrical w.r.t. p1 and p2 then eq. (B) becomes M = 2(Ef
—
~
E1)
~Øf(Pl p2)1J01 (ci ~ p1)J~’(a’~’p~)IØ~(p~ P2)) exp(i~Vwt)exp(—iw’t).
(4)
n
Now since ma’ belongs to veiy weak field we retain only the J0 and J1 functions of a’~’ p1 and cs’~’ p2. We knowJ0(a’~’ p) 1, and J1(a’~’ p) ~-(ci’~’ p). Now by hypothesis the ~ term cannot satisfy energy conservation requirements and it will therefore not contribute. Therefore eq. (A) gives 1~ (E._E.) ~ (Øf(p T~’ 1P2)IJfl(aEp1)Jfl(maEp2)aE p1a’~’ p21c51(p1p2)) (5) and eq. (B) gives 1 (E~ E~) —
T~’
(~f(P1p
2)IJ~,(cia p1)a’~’ p1 I ~i(Pl P2)>.
(6)
In the limit of low intensity eq. (6) becomes
39
Volume 57A, number 1
T~’t= (Ef-- E~)~~t(PiP2)1 ~a
PHYSICS LETTERS
17 May 1976
p 1 a’~’
~~(p1P2)>~
(7)
The method is applicable to multiphoton processes involving arbitrary intensities and frequencies of the incidence radiation under the condition (w/c)a0 ~ 1. We are thankful to Professor SN. Biswas for many valuable discussions and encouragement.
References III
LV. Keldysh, Zh. Eksp. I. Teor. Fiz. 47 (1964) 1945; [Soy. Phys. JETP 20 (1965) 13071; A. Gold and H.B. Bebb, Phys. Rev. Letters 14 (1965) 60. [21 W.C. Renneberger, Phys. Rev. Letters 21(1968) 838. 131 F.I-I.M. laisal, J. Phys. B., Atom Molec. Phys. 6 (1973) L89.
40