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These fluctuations are very small if ii is large. Similarly, the energy fluctuations read
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If the radiated frequencies are strongly peaked about the value <kO), we have
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4-1-3 Induced Absorption and Emission
(4-40)
We can use the explicit forms (4-13) and (4-16) obtained for the S matrix to discuss briefly some interesting situations. Namely, we may wonder how the presence of photons in the initial state affects the radiation of the source j. We have seen in Eq. (4-28) that a classical current creates a coherent state from the vacuum. This coherent state may in turn be considered as the initial state for a second source j. In other words, the first source j;nc' which generates the initial state (4-41) is assumed to be well separated from the second source j(x). The projection of the final state on outstates reads
<b out Ij;nc)
<b inl exp [ - i
d 4 x A;n(x)' j(x) }xp [ - i
d 4 y A;n(Y)' j;nc(Y)] 10 in)
exp [ -
d 4 x d4 y j.(x)G(-)(x - y)jfnc(Y)] d 4 y A;n(Y)' [j;nc(Y)
x <b inl exp
{-i f
+ j(y)]} 10 in)
(4-42)
The first factor in the last expression is a pure phase, independent of the state b, and thus unobservable. The second term tells us that the final state is produced by the sum j;nc + j. The total number ntot of photons is obtained by substituting J + J;nc for J in Eq. (4-26), and the (average) number of radiated photons, defined as the difference ntot - n;nc, is
nmd = ntot -
n;nc
= dk IJ,,(k) 12 + 2 Re dk J,,(k)' Ji':,c.t,(k)
(4-43)
The first term on the right-hand side of (4-43) is the number of photons emitted by the source j alone, while the second one is a typical interference term, which represents the stimulated absorption or emission. We observe that this term is linear in j;nc (or in Ff,;'c) and that terms corresponding to different frequencies are decoupled. This reflects again the independence of the various modes. The reader may compute the radiated energy in the presence of j;nc and observe again this interference phenomenon. On the one hand, we know that two classical sources interfere and energy interferences must imply interferences in photon number since dn(k) = d:i(k)jhko. On the other hand, in view of the stochastic nature of the emission, it is rather surprising to find an interference in the number of emitted photons. This shows clearly the limitation of this stochastic interpretation. Emission and absorption are connected since an emitting source may also absorb photons. Then the equality ntot = n;nc + nj would mean the impossibility of absorption! It is instructive to compare this situation with the case where the initial state has a definite number of photons. For simplicity, we assume all photons to be in the same mode:
INTERACTION WITH AN EXTERNAL FIELD
(4-44)
The function!" is normalized:
and it is understood that f(k) is peaked about a mean value. We shall only compute the average number of photons in the final state
where
Nout
fdk " i...J
.1.= 1,2
d;.)t(k)d;')(k) out out
= Sf
fdk" a().)f(k)a!;')(k)S L.. In In
f dk
f [a!~)f(k) - iJ;.(k)J [a!~)(k) + iJt(k)]
(4-45)
We only count the number of transverse photons, since longitudinal and scalar photons are absent in the initial state and are not emitted. Thus
ntot = n;nc + f dkCIJ , (kJ!2 + IJ2(kJ!2J
+ <n;ncl f dk
id).)f(k)J;.(k)
+ hermitian conjugate ]In;nc)
(4-46)
n;nc + f
dk[IJ , (kW
+ IJ2(kJ!2J
The average number of radiated photons is the same as in the absence of photons in the initial state! On the average, there is no stimulated absorption or emission. But this does not mean that the emission probabilities are unchanged. Indeed, we may compute the probability of finding if! photons in the final state. An easy calculation leads to
Pn,""~rii
= -_--_- ---;In(:
1 an,"" ~ ( exp - { nj an,"" ninc!m! 8z 8z
In(:
.1.= 1,2
Jt(k)f;.(k)
+ complex conjugate
(4-47)
For a weak source J, we may keep the lowest order in J. The only non vanishing transition probabilities to this order are
Pii,""~if;""+1 =
f dk ;.J2IJ;.(kW f dk Sdk
+ n;ncl
S dk
;.~~./;.*(k)J;.(kW
(4-48)
Pn,""~n,"" = 1 Pn,""~n,""-1
;.~~.2 IJ;.(kW ;.~
2n;ncl S dk ;.~~.2 f;.*(k)J;.(kW
1- Pif;""~n,"" -
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