spectrum of visible light) inside the cavity depends on the temperature. We know from experience
with ovens and flames that at moderate temperatures a red glowis visible, while at higher tem-
perature colors move to the yellow, green and blue, namely tohigher frequencies. In the present
section, we shall calculate theabsorption rate of photons in the cavityfrom first principles. The
emission rate may be calculated analogously.
We shall work in a cavity, thus in a spatial box of finite extent, so that photon momenta may
be labeled by discrete vectorsk, and their number can be counted in a discrete manner. The
corresponding radiation oscillators satisfy the canonical commutation relations,
[aα(k),a†β(k′)] =δα,βδk,k′ (20.26)The rates for the different possible values of the wave-vectorkdecouple from one another, and may
be treated separately. Thus, we concentrate on photons witha single value of the wave vectork.
We shall denote by|nα(k);α,k〉the state withnα(k) photons of polarizationαand wave number
k; these states are given by,
|nα(k);α,k〉=1
√
nα(k)!(
a†α(k))nα(k)
|∅〉 (20.27)The numbernα(k) is referred to as theoccupation numberof the state of photon momentumk
and polarizationα.
Absorption will send the atomic state|ψi〉(which is often the ground state of the atomic
part of the system) to an excited atomic state|ψf〉, and will diminish the number of photons.
Considering here the simplest case where absorption occursof a single photon, we shall be interested
in transitions between the following total states,
|i〉= |ψi;nα(k),α,k〉 ≡ |ψi〉⊗|nα(k),αk〉
|f〉=|ψf;nα(k)− 1 ,α,k〉 ≡ |ψf〉⊗|nα(k)− 1 ,αk〉 (20.28)Using again the Fermi golden rule formula for the rate,
Γ = 2π∣∣
∣〈φ|H 1 |i〉∣∣
∣2
δ(Ef−Ei) (20.29)where we use the same approximation for the photon-matter coupling that we used for the single
photon emission calculation,
H 1 =e
m
A·p (20.30)where the electro-magnetic fieldAis in the transverse gauge. The first part of the calculation
consists again in evaluating the matrix elements of theA-field, but this time between two states
with multiple photons. To evaluate it, we need the followingrelation,
aβ(k′)|nα(k),α,k〉= (nα(k))(^12)
|nα(k)− 1 ,α,k〉δα,βδ(3)(k′−bk) (20.31)