Chapter II: Strong atom-light interactions: The mostly classical story
2.3 The power radiated by an oscillating dipole
An oscillating dipole radiates energy into the environment, and the amount of power radiated is in turn influenced by the environment. Systems can be designed to either enhance or suppress the dipole radiation and thereby decrease or increase the decay rate. In this section, we will see how the decay rate of the dipole energy is related to the Green’s function.
The time averaged power radiated out of a systemPradcan calculated by performing a surface integral of the normal component of the Poynting vector over a surface surrounding the oscillating dipole. It can also be calculated by integrating the line- currentj(r)with the in-phase electric field, which performs work on the line-current,
Prad =−1 2
∫
d3rRe[j(r) ·E(r, ω)]. (2.20)
An oscillating dipole moment at position r0 results in the line-current j(r) =
−iωpδ(r−r0), and therefore the power radiated is Prad = ω
2 Im[p∗·E(r0, ω)]. (2.21) Using Dyson’s equation, we replace the electric field at the dipoleE(r0)with the self-Green’s function,E(r0, ω)= µ0ω2G(r0,r0, ω) ·p, which then gives
Prad = ω3|p|2
20c2 nˆp·Im[G(r0,r0, ω)] ·nˆp. (2.22) Here, ˆnp is the unit vector of the dipole, p = |p|nˆp . Note that we have assumed that ˆnp is real, e.g. for a linear dipole moment. In order to convert the power radiated into a decay rate, we must also know the initial total energyW(t=0)of the dipole. The averaged energy of a dipole is the sum of the averages of the kinetic and potential energy,W = hmxÛ2/2+ k x2/2i = h m
2q2(ω20p2(t)+pÛ2(t))i = mω2q202|p0|2, whereqis the charge of the oscillator andmis the mass. Assuming that the energy decays exponentially asW(t) =W(0)e−Γt, the power radiated is related to the total energy and decay rate by Prad(0)= dtdW(t)|t=0 = −ΓW(t=0). Solving for the decay rate and substituting the energyW and powerPradgives
Γ= 2µ0ω2 q2
2mω nˆp·Im[G(r0,r0, ω)] ·nˆp. (2.23) The decay rate of an oscillating dipole is proportional to the imaginary part of the self-Green’s functionG(r0,r0, ω)at the position of the dipole. The decay rate of a dipole can be enhanced or inhibited by the environment.
We will often want to take the ratio of the decay rate in a system to the decay rate in free-space, a ratio which is known as the Purcell factor [47]. To find the free-space Green’s function at the source (r = 0), we perform a Taylor series of the electric field in Eq. (2.17) and (2.18) over small kr. Keeping both the first order real and imaginary terms, the result is
kr→lim0E(r, ω)= µ0ω2|p| ω 4πc
2
(kr)3 −i2 3
z.ˆ (2.24)
The electric field is parallel to the dipole orientation. The real part is the same as the DC dipole case and diverges atr= 0. The divergence is a result of the dipole being infinitesimally small and would go away if we considered instead an oscillating polarization density for a sphere of any radius. In contrast, the imaginary part of
the electric field, which is out-of-phase with the oscillating dipole, is finite. Using Dyson’s equations, the imaginary part of the self-Green’s function is
Im[G0(r0,r0, ω)]= ω 6πc
↔I. (2.25)
Inserting this result into the decay rate gives the classical expression for the free- space decay rate,
Γ0= µ0ω2 q2 2mω
ω
3c, (2.26)
which depends on the particle’s chargeqand massm. The Purcell factor - the ratio between the decay rate and the free-space decay rate- is then
PF=Γ/Γ0= 6c
ω nˆp·Im[G(r0,r0, ω)] ·nˆp. (2.27) The Purcell factor is independent of the charge and mass, and is also valid for the Purcell factor of a two level system in the quantum derivation (see Sec. 3).
In the quantum theory for an atom, the free-space decay rate of a non-degenerate excited state|eito a ground state|giis instead
Γ0= µ0ω20
~
ω0|hg|pˆ|ei|2 3πc
, (2.28)
which can be calculated using the Wigner-Weisskopf approximation [48]. The quantum result can be obtained from our classical result by replacing the dipole energy with ~ω0, and by replacing the dipole moment with half of the dipole moment matrix element, p → hg|pˆ|ei/2. The factor of 1/2 is due to the fact that the Fourier transform of the classical dipole moment spans positive and negative frequencies.
Optical cross-section.
Another way to characterize the power radiated by a dipole is in terms of the optical cross sectionσ, which can be interpreted as the source area for an incoming plane wave that is scattered. It is calculated by dividing the power radiated over the incoming intensity Iin, σ = Prad/Iin. Remarkably, the on-resonant optical cross- section for an oscillating dipole is independent of the charge and mass, and is given by
σ0= 3λ2
2π . (2.29)
This result only requires that the decay of the dipole oscillator is due only to radiated power, or equivalently to the radiative back-action caused by resulting field. In the quantum formalism, this result will also be true for the on-resonant optical cross- section of a two level system.
The proof of the optical cross-section is the following. The intensity for an incoming plane wave of amplitude E0 is given by Iin = c20|E0|2. The power radiated (see Eq. (2.21)) is Prad = ω2Im[p∗ · E(r0, ω)], which after substituting polarizability equationp= α(ω)E(r0, ω), gives
Prad= ω
2|E(r0, ω)|2Im[α(ω)].
The optical cross-section is defined as the ratio of the radiated power to the incoming intensity, σ = Prad/Iin. Under the Born approximation, E(r, ω) ≈ E0, in which case σ(ω) = ω20Im[α(ω)]. From Eq. (2.32), the on-resonant polarizability for a dipole is α(ω0) = iq2/mω0Γ0, and then the on-resonant optical cross-section is σ0 =q2/m0cΓ0. After substituting the decay rate from Eq. (2.26) into this equation, the charge and mass terms cancel, and we getσ0 = 3λ2π2.