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Response of an absorber to resonant radiation
&xnical
Physics 8 (197%
399-404
Publishing Compsny
0 North-Hound
RESPONSE OF AN AUSOREER TO RESONANT RADIATION
1. Introduction Several semi&ssical
studies [I -81 of the responses of atom5 or collections of atoms, obeying the optical
Bloch equations [9,iO] 1to are
incomplete
pukes
oi
in the sense ihzt the
resonant or near resonxu radiation have recently appeared. These “theor2ticsl”
treatments
atoms manifest no damping ofanp kind, in contrast IO the bc-
haviar of real atoms. On thlt other hand hlowcr [I 11Ins pruvided 3 fullyquantum meehatical formulation for the dcosy of prepared states of the joint atom-radhtion system. in which (radiative) decay of the atomic exated state has b?en included. (Indeed, the decsy of both nrigirlatirtg and termm~ting atomic states fnr the optical pumprng are allowed IO decay in blower’s Irealmenr.) Huwvrvsr. in is nix rmmediateiy c\ldent JUSI h~\r: MOBW’S rtsults
theories. ~41ich arc generally exprcsscd in lcrms of the in. 3nd out-of-phssr art related to those of the semiilusacal parts oi the atomtc rranstticwdlpole rcsponw. It is also not readily spparcnt how to itrclude the shape of that Ii@ pulse ia any conventional treatment of the decay of a prepwd state, since the irrewrsible decay itself inherently places early nnd law timcs on an unequal fuotmg. Although these problems in relstq the drcay-af-preparcdstates method to the semiclsssical theories appear to be rather difficult to rrezrt in general. for IZSOKIIIC~ radiation an appreciable simplifiL>tion occurs that allows some insight to be gained. In addition a general prescription a~-
plicabte to square pulses of light can be formulated for obt3inin~ the out.of=phase part of the atomic transition* dipole response.
2. Probability smplitude of the initial stttte “f. IOgive rhc probahiltty ampilitude f,(r) for hlowcr’s treatment [I I J is readily extended, using his equ3tlon (-the initial state ia) = &$\~~),r~),consisting of ground state absorber and II photons in the radiation mode with wewtor
X and polariwtion I,(t)
where
v. thus
= [e-iaf-ef/2(btid)]
{e -%?dr [a tb-&
f i(-CtdSr~,
‘?_)) +&rc*dt i -o+b fi?b + i(c+d-rbbt?)]),
(1 )
(3~‘cos(R’r~sin(R’r)+(~b.l3)jtn~(~’~)~ 7-h - (rbww) --~ .~ -__I_) __ __.________ _ u(r)=@ [cos(R’r)t(fblJR’)sln(R’r)JZ -t \Qlg,!(R’)2]m~(R’r)
w&h is e~~entj~i]y identical to the result of McCall and Hahn for 4 pulse of resonant r~dl~iion.The missing factor of:! in the argument of the sine jn the present case arises from Ihe consideration of line& rtilher than ~‘irc&rjp polarized light. If there ate two orthoSon~1 ~on~ponents for E(t) =E,(Oi +E,(t)i. then (mot *E) becomes = ztft, when /m,f = ~m.,.f= m. When also mrtximumamplitudes of the orthogonsl eiectric field n++NtY*j components are the same (i.e., CircuIar as opposed to ellipticsl pofxization) and equal to /El, then the relation between (i(i) and u(f) becomes (IV) = f u(r)lEjwl or twice the Me Of eflergy triInsfer ds in eq. (8), so that for circuhrly polarized light of amplitude IEI there results
which is precisely the expression (1s) of hlccall
and Hahn. me agreemenobtainedherebetween the decay of prepared states tr~tntent and the result of h!cCalland
Hati for the specisicaseof no damping portends well for the validity of eq. (14) for square
pulses of
resonant
light even in the presence of damping. The question thewcalled adiabatic followin_asolution of Grischkowsky 13.61 and coworkers [4.5], recent]!
of
rigorously derived from the optical Bloch equations by Crisp IS]. and its relation to the decay of prepxed states method is not considered here. but will be subsequently discussed elsewhere I12,13].
Appendix:
Response to 3 resonant
pulse of time-varyingintensity in the zero-dampinglimit
The pulse is divided up into 3 series of intervals O+ I 1, II + rz, etc.. of duration 61 E t,+ 1- fi, ow any one of assensibly eo~ls~~t~t. At [heSW Oftheith InterViI~t**t- 1, the ww I-tmion
which the pul~e may btt regarded pqrp
=d(t
f
)ld
‘I
i+
f b(f$fb).
is
(At
and at the end of that same interval the wave function
1
is
16(r))=19(ri+Si))=d(ijfSi)la)il;(ri+6i)lb!.
(A21
The coefficients in cq. (AZ) are readily found from the relations “(5 + +I = f&g
8l/) f G&I
6($
(A31
&t+&t, =It$J(61)b(~~+&&,~ dir,).
Irt-l)
where the subscripts j in the orgumenrs of the probability
amplitudes indicate not only the time interval
itsdf, but
also specify that it is the li$tt intensity $ which is applicablein computing that probability amplitude. In the limit of zerq-damping at resonancr the following values prevail during the jth interval: r-b = 0, (Yb)i
&F~=O. =
-i(Qlj,@
-4 = “Q’i,
B=O,
Q=Q$),
bi=~ (Qri)“‘,
=(H&),..
c=d=@. (AS)
with these substitutions it is read& found from eq. (1) and its corresponding analogue for ibb{8:t) that I,@jf
= Wp (-ia6ikos(bi&if
Frm hfwm’s &(6j)
Thus,
=
= &,,($).
ssprzssion(37)for &,(r)and its correspondin\S analoguc: exp (-iu$) sin(b+$) = --fbb($),
(A3) and (Al) bxome
(:\6} for 1&f
there results 31~0 (A71
J.M. SAurlRcsponrc of at!absorberIOtesollanlradralion
b(fj *6j)
=eXP
(-lf7dj)
lb($)COs(b,di) -6rIj)sin(b,JiJ].
(A9)
expressions for the end of the (j+l)th interval can be obtained from (A8)and Since the start ofthe(j+l)th interval occurs at I~+, = fitSi. there results
Corresponding j+jtl.
(A9) by setting
d(rj+~jf6/+t)=esP(-in~,+*)]ri(~+l5,)c~S(b,+t&,+~)+D(~t6j)Sin(b,+td,+t)]
=exP
[-iQ(~j+~j+t)I
[~(ri)Cos(bj~jtbj+t6i+t)
-rjflj)sin(b$,
tbj+t6,+t)].
(Al I )
If WC now set I’=O. (or Ii =O). and apply repeatedly the cycle of arguments leading from (A8) and (A9) to (AIO) and Al I), or use proof by induction.
there results for T= I,$+:
~~~)=e~~(-i~~~j)(~~O,~~~(~bj~~)id(0)sin(~~;~j)),
(Al 3)
(A13)
Ii the initial condition is IWO)) = la), 3s assumed here. then using b, = (Q!,i1/2 = lmol -l? IIE,I/s. whcrc 151 is the ~4cctric field swnngrh of the pulse during the jth interval, one obtains, upon converting the sums to integrals, rhc following d(T)
expressions.
= e-iaTcos
(Al4i
f+llh(i),dr). 0
b(T)= -e-iaTsin ((~j,,,,d~). Vs~ng eq. (IO) for the ourlofmphasc place of IEo() there results u(T) = (mO,
l
(AIS)
component of the transition
d~pol~l ~notnent al
tinrc! 7’6 e.,
E) sin fvj,E(!,,dI).
ivhich should be valid for pulses of resonant light that bury slowly in time compared IO cos WII.
References [I] S.L. McCall and E.L. Hahn, Phys. Rev. Lerrers I8 (1967) 906. 12) S.L. S~CC;IUand E.L. Hahn, Phys. RCV. 183 (196% 457. I;] D. Grischkowsk)‘, Phys, Rev. Lclters 21 (1970) 8668 iJ\ D. Gnschkowsky and J.A. Armsrrong, Phgs. Rw. A6 (19721 1566. 151 D. Crischkowsky. Phys. Rev. A6 (1973) 2096. 161 D. Crischkowsky. E. Courtens and J.A. Armslronf, Phys. Rev. LctKrs 31 (1973) 421.
dh
I./Ml k (Al@
(71 M.D. Crisp. Phys. Rev. l_~~rrrrs22 (1969)810. I81 M.D. Crisp, Ploys. Rev. A8 (!97?) 7178 (91 R.P. frynman. T.L. scrnon. JI. and R.W. tldlwrth. J. t\P@. (101 F. Bloch, Phys. Rev. 70 (19-%)460. 11 i ! L. Mower, PhyT. Rw. l-12 t I 966) 799. (I$_] j.hf. Schurr. manuscrrpt rn pWpJfJtiOR. (131 J.M. Srhun. Inrern. J Quantum Ckni. 5 11971) 15.
Phfs.
3s 11957119.
Physics 8 (197%
399-404
Publishing Compsny
0 North-Hound
RESPONSE OF AN AUSOREER TO RESONANT RADIATION
1. Introduction Several semi&ssical
studies [I -81 of the responses of atom5 or collections of atoms, obeying the optical
Bloch equations [9,iO] 1to are
incomplete
pukes
oi
in the sense ihzt the
resonant or near resonxu radiation have recently appeared. These “theor2ticsl”
treatments
atoms manifest no damping ofanp kind, in contrast IO the bc-
haviar of real atoms. On thlt other hand hlowcr [I 11Ins pruvided 3 fullyquantum meehatical formulation for the dcosy of prepared states of the joint atom-radhtion system. in which (radiative) decay of the atomic exated state has b?en included. (Indeed, the decsy of both nrigirlatirtg and termm~ting atomic states fnr the optical pumprng are allowed IO decay in blower’s Irealmenr.) Huwvrvsr. in is nix rmmediateiy c\ldent JUSI h~\r: MOBW’S rtsults
theories. ~41ich arc generally exprcsscd in lcrms of the in. 3nd out-of-phssr art related to those of the semiilusacal parts oi the atomtc rranstticwdlpole rcsponw. It is also not readily spparcnt how to itrclude the shape of that Ii@ pulse ia any conventional treatment of the decay of a prepwd state, since the irrewrsible decay itself inherently places early nnd law timcs on an unequal fuotmg. Although these problems in relstq the drcay-af-preparcdstates method to the semiclsssical theories appear to be rather difficult to rrezrt in general. for IZSOKIIIC~ radiation an appreciable simplifiL>tion occurs that allows some insight to be gained. In addition a general prescription a~-
plicabte to square pulses of light can be formulated for obt3inin~ the out.of=phase part of the atomic transition* dipole response.
2. Probability smplitude of the initial stttte “f. IOgive rhc probahiltty ampilitude f,(r) for hlowcr’s treatment [I I J is readily extended, using his equ3tlon (-the initial state ia) = &$\~~),r~),consisting of ground state absorber and II photons in the radiation mode with wewtor
X and polariwtion I,(t)
where
v. thus
= [e-iaf-ef/2(btid)]
{e -%?dr [a tb-&
f i(-CtdSr~,
‘?_)) +&rc*dt i -o+b fi?b + i(c+d-rbbt?)]),
(1 )
(3~‘cos(R’r~sin(R’r)+(~b.l3)jtn~(~’~)~ 7-h - (rbww) --~ .~ -__I_) __ __.________ _ u(r)=@ [cos(R’r)t(fblJR’)sln(R’r)JZ -t \Qlg,!(R’)2]m~(R’r)
w&h is e~~entj~i]y identical to the result of McCall and Hahn for 4 pulse of resonant r~dl~iion.The missing factor of:! in the argument of the sine jn the present case arises from Ihe consideration of line& rtilher than ~‘irc&rjp polarized light. If there ate two orthoSon~1 ~on~ponents for E(t) =E,(Oi +E,(t)i. then (mot *E) becomes = ztft, when /m,f = ~m.,.f= m. When also mrtximumamplitudes of the orthogonsl eiectric field n++NtY*j components are the same (i.e., CircuIar as opposed to ellipticsl pofxization) and equal to /El, then the relation between (i(i) and u(f) becomes (IV) = f u(r)lEjwl or twice the Me Of eflergy triInsfer ds in eq. (8), so that for circuhrly polarized light of amplitude IEI there results
which is precisely the expression (1s) of hlccall
and Hahn. me agreemenobtainedherebetween the decay of prepared states tr~tntent and the result of h!cCalland
Hati for the specisicaseof no damping portends well for the validity of eq. (14) for square
pulses of
resonant
light even in the presence of damping. The question thewcalled adiabatic followin_asolution of Grischkowsky 13.61 and coworkers [4.5], recent]!
of
rigorously derived from the optical Bloch equations by Crisp IS]. and its relation to the decay of prepxed states method is not considered here. but will be subsequently discussed elsewhere I12,13].
Appendix:
Response to 3 resonant
pulse of time-varyingintensity in the zero-dampinglimit
The pulse is divided up into 3 series of intervals O+ I 1, II + rz, etc.. of duration 61 E t,+ 1- fi, ow any one of assensibly eo~ls~~t~t. At [heSW Oftheith InterViI~t**t- 1, the ww I-tmion
which the pul~e may btt regarded pqrp
=d(t
f
)ld
‘I
i+
f b(f$fb).
is
(At
and at the end of that same interval the wave function
1
is
16(r))=19(ri+Si))=d(ijfSi)la)il;(ri+6i)lb!.
(A21
The coefficients in cq. (AZ) are readily found from the relations “(5 + +I = f&g
8l/) f G&I
6($
(A31
&t+&t, =It$J(61)b(~~+&&,~ dir,).
Irt-l)
where the subscripts j in the orgumenrs of the probability
amplitudes indicate not only the time interval
itsdf, but
also specify that it is the li$tt intensity $ which is applicablein computing that probability amplitude. In the limit of zerq-damping at resonancr the following values prevail during the jth interval: r-b = 0, (Yb)i
&F~=O. =
-i(Qlj,@
-4 = “Q’i,
B=O,
Q=Q$),
bi=~ (Qri)“‘,
=(H&),..
c=d=@. (AS)
with these substitutions it is read& found from eq. (1) and its corresponding analogue for ibb{8:t) that I,@jf
= Wp (-ia6ikos(bi&if
Frm hfwm’s &(6j)
Thus,
=
= &,,($).
ssprzssion(37)for &,(r)and its correspondin\S analoguc: exp (-iu$) sin(b+$) = --fbb($),
(A3) and (Al) bxome
(:\6} for 1&f
there results 31~0 (A71
J.M. SAurlRcsponrc of at!absorberIOtesollanlradralion
b(fj *6j)
=eXP
(-lf7dj)
lb($)COs(b,di) -6rIj)sin(b,JiJ].
(A9)
expressions for the end of the (j+l)th interval can be obtained from (A8)and Since the start ofthe(j+l)th interval occurs at I~+, = fitSi. there results
Corresponding j+jtl.
(A9) by setting
d(rj+~jf6/+t)=esP(-in~,+*)]ri(~+l5,)c~S(b,+t&,+~)+D(~t6j)Sin(b,+td,+t)]
=exP
[-iQ(~j+~j+t)I
[~(ri)Cos(bj~jtbj+t6i+t)
-rjflj)sin(b$,
tbj+t6,+t)].
(Al I )
If WC now set I’=O. (or Ii =O). and apply repeatedly the cycle of arguments leading from (A8) and (A9) to (AIO) and Al I), or use proof by induction.
there results for T= I,$+:
~~~)=e~~(-i~~~j)(~~O,~~~(~bj~~)id(0)sin(~~;~j)),
(Al 3)
(A13)
Ii the initial condition is IWO)) = la), 3s assumed here. then using b, = (Q!,i1/2 = lmol -l? IIE,I/s. whcrc 151 is the ~4cctric field swnngrh of the pulse during the jth interval, one obtains, upon converting the sums to integrals, rhc following d(T)
expressions.
= e-iaTcos
(Al4i
f+llh(i),dr). 0
b(T)= -e-iaTsin ((~j,,,,d~). Vs~ng eq. (IO) for the ourlofmphasc place of IEo() there results u(T) = (mO,
l
(AIS)
component of the transition
d~pol~l ~notnent al
tinrc! 7’6 e.,
E) sin fvj,E(!,,dI).
ivhich should be valid for pulses of resonant light that bury slowly in time compared IO cos WII.
References [I] S.L. McCall and E.L. Hahn, Phys. Rev. Lerrers I8 (1967) 906. 12) S.L. S~CC;IUand E.L. Hahn, Phys. RCV. 183 (196% 457. I;] D. Grischkowsk)‘, Phys, Rev. Lclters 21 (1970) 8668 iJ\ D. Gnschkowsky and J.A. Armsrrong, Phgs. Rw. A6 (19721 1566. 151 D. Crischkowsky. Phys. Rev. A6 (1973) 2096. 161 D. Crischkowsky. E. Courtens and J.A. Armslronf, Phys. Rev. LctKrs 31 (1973) 421.
dh
I./Ml k (Al@
(71 M.D. Crisp. Phys. Rev. l_~~rrrrs22 (1969)810. I81 M.D. Crisp, Ploys. Rev. A8 (!97?) 7178 (91 R.P. frynman. T.L. scrnon. JI. and R.W. tldlwrth. J. t\P@. (101 F. Bloch, Phys. Rev. 70 (19-%)460. 11 i ! L. Mower, PhyT. Rw. l-12 t I 966) 799. (I$_] j.hf. Schurr. manuscrrpt rn pWpJfJtiOR. (131 J.M. Srhun. Inrern. J Quantum Ckni. 5 11971) 15.
Phfs.
3s 11957119.