2.4 Atomic Transitions
2.4.1 Transition Probabilities
A good starting point for atomic transitions is to consider the absorption and emis- sion probabilities of a two level atom with ground state i and excited state j, as shown in Fig. 2.3, in a radiation field of angular frequency density ρ(ω). The transition frequency ωji is given by
~ωji =Ej−Ei (2.24)
where Ej and Ei are the energies of the two states (by definition Ej > Ei). There are three processes to consider: spontaneous emission, absorption and stimulated
emission, with associated probabilities A, Bij and Bji - the Einstein coefficients [80]. If the state populations at time t are Ni and Nj, then the rate of change of
Ni and Nj are proportional to the three Einstein coefficients times the appropriate state population, times the energy density per unit frequency range (ρ(ω)) for the non-spontaneous processes. This leads to the rate equation
−dNj dt =
dNi
dt =ANj −Bijρ(ω)Ni+Bjiρ(ω)Nj. (2.25)
With no applied radiation – i.e. the case of spontaneous emission which we are interested in – the solution to Eqn. 2.25 becomes
N2(t) = N2(0)e−At. (2.26)
From this we can see that the spontaneous transition lifetime τ = 1/A.
The treatment can also be generalised if there are multiple levelsito which decay is possible, via the individual decay probabilitiesAji. Eqn. 2.25 will then be
− ddNtj =X i
AjiNj (2.27)
leading to the total decay lifetime τj = 1/(PiAji).
To determine the value of the Einstein coefficients there are two textbook ap- proaches used to treat the problem of the interaction between an atom and a weak radiation field using time-dependent perturbation theory. The first approach is to quantise the atomic energy levels but treat the radiation field classically, a semi- classical treatment. However, while this approach is adequate to explain induced emission and absorption processes, it breaks down somewhat when attempting to describe spontaneous emission. This is because it relies on the classical assumption that the number of radiation modes ¯n≫1, which is clearly not true for spontaneous emission. To provide a correct treatment of this process the radiation field must also be quantised, using quantum electro-dynamics (QED). Such an approach is beyond the scope of most introductory texts on the subject [80, 55] and given some physical insight can be provided in a much simpler manner using the semi-classical treatment, this is all the detail that will be provided in this thesis.
In order to accomplish this we must use the relationship between the three Ein- stein coefficients which can be derived in a relatively straightforward manner [80] but is merely stated here
Aji = ω2 ij π2c3~ωijBji = ω2 ij π2c3~ωij gi gj Bij (2.28)
where the levels i, j are gi-fold andgj-fold degenerate.
The Einstein coefficients are determined in the semi-classical approximation by taking the Schr¨odinger equation of the electron
i~dΨ
§2.4 Atomic Transitions 19
and modifying the Hamiltonian (H) of the electron to include the unperturbed component (H0) and a time dependent perturbation due to the (classical) radiation field (H′(t)). In the weak interaction limit [80] we can write
H′(t) = e
mA·p (2.30)
where e and p are the electron’s charge and momentum respectively while
A=A0ˆe ei(ωt−k·r)+e−i(ωt−k·r)
(2.31) is the electromagnetic vector potential of the radiation field, which is assumed to be an incoherent superposition of plane waves polarised in the direction of the unit vectorˆe.
The electron wavefunction in Eqn. 2.29 is then re-written in terms of stationary (time-independent) states ψn of the atom such that
Ψ = X n
cnψne−iEnt/
~
(2.32) where individual ψn are space-dependent wavefunctions which are solutions of the unperturbed Hamiltonian
H0ψn =Enψn. (2.33)
The physical interpretation of the time-dependent coefficients cn is that if the atom is in state i at time t = 0 (i.e. ci(0) = 1, cn6=i(0) = 0), then the probability of finding the atom in statej at timet is|cj(t)|2. Thus we can find the transition rate |cj(t)|2/t by solving the differential equation (DE)
X n (i~c˙n+Encn)ψnexp −iEnt ~ = (H0+H′(t)) X n cnψnexp −iEnt ~ (2.34) which results from substituting Eqn. 2.32 into Eqn. 2.29 ( ˙cn represents the time derivative of dcj(t)/dt). However, because ψn is an eigenfunction of H0 with eigen- value En, the second term in the left hand side (LHS) of Eqn. 2.34 cancels with the first term on the right hand side (RHS). Under the additional assumption that
H′(t) is small enough thatc
ndoes not change significantly over time we can use the initial values of cn (i.e. ci(0) = 1) for the RHS of Eqn. 2.34. These approximations give X n i~c˙nψnexp −iEnt ~ =H′(t)ψ iexp −iEit ~ . (2.35) By multiplying by ψ∗
j, integrating over spatial co-ordinates and re-arranging we obtain i~c˙j =hj|H′(t)|iiExp i(Ej−Ei)t ~ (2.36) because hm|ni= 0 for m6=n, so the off diagonal terms on the LHS will be zero.
tonian given in Eqn. 2.30, with the radiation field given by Eqn. 2.31. Further simplification is attained using the frequency-energy level relation in Eqn. 2.24, so that the DE becomes
i~c˙j =hj| e mˆe·pe −ik·r|iiA 0 ei(ωji+ω)t+ei(ωji−ω)t . (2.37)
This can be solved using the initial condition cj(0) = 0. If we consider the case of absorption (Ej > Ei), the solution for the transition probability Pab = |cj(t)|2/t under the rotating wave approximation (ω ≈ωji) then becomes
Pab = hj| e mˆe·pe −ik·r |ii 2 A20sin 2((ω ji+ω)/2) ~2((ω ji+ω)/2)2 . (2.38)
Integrating the RHS over all incident radiation frequencies ω in the limit of large t
we obtain the radiation densityρ(ω), so that
Pab = π ǫ0~2ω2ij hj| e mˆe·pe −ik·r |ii 2 ρ(ωji). (2.39) From Eqn. 2.25 it is clear that Pab = Bijρ(ωji), so Eqn. 2.39 yields the Einstein coefficient of absorption Bij directly. The only further calculation required is to determine the matrix element hj|e
mˆe·pe
−ik·r|ii
2
connecting states j and i.