Page 157 - Principles and Applications of NanoMEMS Physics

3. NANOMEMS PHYSICS: Quantum Wave Phenomena 145

3.2.3 Cavity Quantum Electrodynamics
The field of Cavity Quantum Electrodynamics or, cavity QED, deals with
the effect of the surrounding environment on the spontaneous emission rate
of atoms [171]. The concept was introduced by Purcell in 1946 [171] in the
context of nuclear magnetic moment transitions. He observed that at
conditions of temperature, radio frequency, and nuclear magneton given by
−
300°K, ν = 10 7 sec , and µ = 1, respectively, the corresponding rate of
1
spontaneous emission, given by,

§ 8πν 2 · § 8π 3 µ 2 ·
A = ¨ ¨ 2 ¸ ¸h ¨ ν ¨ 2 ¸ ¸ sec − 1 , (201)
ν
© c ¹ © 3h ¹
−
adopts a value of 2× 10 − 22 sec . So small is, indeed, this value, that it
1
implies the virtual impossibility of the spin system being able to achieve
thermal equilibrium with its surroundings. This expression, Eq. (201 ), for the
spontaneous emission rate A between initial and final states i and f ,
assumes the atom is in free space and derives from Fermi’s golden rule
[172], namely,

2
f H i
A = ρ () ν , (202)
= 2

where the initial state i , represents an atom in the absence of any photons,

and the final state f , represents the atom with a single photon. The
Hamiltonian H represents the atom-field interaction, and () νρ represents

the density of photon states or number of radiation oscillators per unit
volume, in a unit frequency range which, for free space, adopts the value of ,

ρ = (8πν 2 c 3 ). (203)
S

In other words, ρ embodies the number of electromagnetic modes into
S
which photons may be emittted at the location of the emitter [173].
When the atom is enclosed by a microwave cavity of quality factor Q,
however, the number of radiation oscillators per unit volume is limited to
those occupying the frequency range ν Q , which is, in fact, exactly one. If

one assumes the cavity volume and the wavelength to be related by