At optical frequencies spontaneous emission often plays the dominant role in the dynamics of atomic systems. As shown previously, the simple two-level atom inter- acting with a single field mode is unable to account for the experimentally observed decay. The first successful method of explaining spontaneous emission was proposed by Weisskopf and Wigner in 1930 [5]. Following their method we show that by in- cluding the coupling to a continuum of free-space electromagnetic field modes the
2.4. WEISSKOPF-WIGNER THEORY 23
rate of decay can be deduced.
The Hamiltonian of a two-level atom interacting with a infinite set of field modes is given by H =~ω1σ11+~ω2σ22+~X s ωsˆa†saˆs+~ X s gs(σ+ˆas+σ−ˆa†s), (2.18)
where we have removed the zero-point energy associated with each field mode. We now restrict the atom to the situation where only one photon is present in one of the field modes while the atom is in the ground state. The restricted basis can be written as
|si = |1iA⊗ |. . . ,0,1,0, . . .i, (2.19)
|ei = |2iA⊗ |0,0, . . . ,0i, (2.20)
where s ∈ [0,∞) labels which of the infinite set of field modes the photon is in. Within this basis the Hamiltonian becomes
H =~ω2σee+~X s (ω1+ωs)σss+~ X s gs(σs++σs−), (2.21) where σs+ = |eihs|. This Hamiltonian describes an excited state, |ei, coupled to a infinite set of lower levels, |si. The set of energy levels |si represent the atom in the ground state with one photon of frequency ωs. Thus, we have transformed our
Hamiltonian into the form of a classical photoionisation problem. To take account of the infinite set of field modes we change the summation into a three dimensional integral over the density of states:
X
s
−→
Z
D(ωs)d3ωs. (2.22)
The integral is taken over all possible field modes. Here D(ωs) is the density of
states, which in free space is frequency independent, isotropic and is found to be
D(0) = 2V /(2πc)3. It is important to recall that the dipole coupling element is a function of both the frequency and orientation of each field mode with respect to the atomic dipole moment. That is
g(ωs) = p·ˆeωsωs
24 CHAPTER 2. INTRODUCTION TO QUANTUM ELECTRONICS
where θ is the angle between the atomic dipole and the polarisation of the field mode ωs. Transforming into an interaction picture we get the form of a Hamiltonian
describing the coupling between a single bound state and an isotropic continuum of free space modes:
H =~ Z ∆sσssD(0)d3ωs+~ Z g(ωs)(σs++σs −)D(0)d3ωs, (2.24)
where ∆s = ω1 −ω2 +ωs. We assume that the solution to the dynamics is of the
form
|ψ(t)i=ce(t)|ei+
Z
cs(t)|siD(0)d3ωs. (2.25)
This results in the infinite set of coupled equations for the time-dependent coeffi- cients: ˙ ce(t) = −i Z g(ωs)cs(t)D(0)dωs, (2.26) ˙ cs(t) = −i∆scs(t)−ig(ωs)ce(t). (2.27)
We now formally integrate the equation (2.27) and substitute this into the differential equation (2.26). This transforms the two coupled differential equations into a single integro-differential equation for the coefficient ce(t). Once the angular integrations
have been performed we find
˙ ce(t) = − p2 6π2c3~ 0 Z ∞ 0 dωsω3s Z t 0 dt0ce(t0)ei∆s(t 0−t) . (2.28)
So far this equation is exact. However, we now note that for large values of ∆s
the time integral makes a vanishing contribution, varying approximately as ∝1/∆s.
Since only frequencies around resonance contribute significantly, we can make the approximation ω3 s =ω213 . Therefore ˙ ce(t) = − p2ω3 21 6π2c3~ 0 Z ∞ ω1−ω2 d∆s Z t 0 dt0ce(t0)ei∆s(t 0−t) . (2.29)
When the integral over the detunings is evaluated we obtain Z ∞
ω1−ω2
d∆sei∆s(t
0−t)
=πδ(t0−t) +iP. (2.30) The term P is a principle value integral that leads to an energy shift of the state
2.4. WEISSKOPF-WIGNER THEORY 25
with non-zero Rabi-frequency and is closely related to the Lamb shift in hydrogen [6, 17]. Normally the energy shift is very small and is absorbed into the definition of the natural frequency ω21. In fact, this is an elementary example of the method of renormalisation often used in quantum field theory. Evaluating the time integral we find that the decay rate of the excited state is
˙
ce(t) = −
p2ω3 21
6πc3~0ce(t). (2.31)
This is easily solved to give the observed exponential decay of the excited atomic state:
ce(t) = exp(−Γt/2)ce(0), (2.32)
where the spontaneous decay rate has the value
Γ = p 2ω3
21 3π0~c3
. (2.33)
We note that the spontaneous decay rate of the excited state is proportional to the cube of the transition frequency. This explains why decay rates of the order 10Hz are possible at microwave frequencies, as opposed to 10MHz at optical frequencies. Spontaneous emission can also be reduced by decreasing the number of field modes present, as is often done by placing the atom in a high Q-factor cavity.
Commonly spontaneous emission is explained as arising due to stimulated emis- sion of the atom by the vacuum field modes. It should be noted however, that the calculation above makes no direct reference to the zero-point fluctuations of the vacuum fields. These fluctuations are in fact neglected at the very beginning of the calculation. Indeed, were this explanation complete one would expect spontaneous absorption of vacuum fluctuations to occur also, contrary to experimental evidence. In has therefore been suggested that a more classical interpretation of spontaneous emission should be employed [17]. For instance, when in an excited state the atom’s own non-vanishing electromagnetic field (Ω 6= 0) should be viewed as causing a ra- diative reaction that results in decay. This explanation is directly analogous to the classical Lorentzian theory of radiative decay.
26 CHAPTER 2. INTRODUCTION TO QUANTUM ELECTRONICS