Quantum Mechanical Background

Quantum Mechanical Background

amplitude, which then has ?uctuations. These ?uctuations are called vacuum ?uctuations, because they exist even in the vacuum, i.e., even when no clas- sical ?eld exists. The vacuum ?eld has a continuous spectrum. When this is used with the quantized ?eld version of (3.78) (see Sect. 14.3), we ?nd the
spontaneous emission rate constant called A by Einstein and ? in (3.89). To ?nd the value of A intuitively, we can use the rate given by (3.117) if we can guess what the energy density U (?) of the vacuum ?eld is. We note that the number of ?eld modes per unit volume between ? and ? + d? is ?2/?2c3 [see (14.46)]. Multiplying this number by the energy k? of one photon, we have Uspon(?) = k?3/?2c3. Using this in (3.117) gives the spontaneous emission
rate

A = B k?

= ? ?

. (3.118)

?2c3

3??0kc3

The lifetime of the upper level is 1/A. This is the same result (14.60) as
derived in detail in Sect. 14.3. This section also shows that spontaneous
absorption does not occur.
We can use these facts to derive informally the Planck blackbody spec- trum

k?3/?2c3
U (?) = ek?/?

, (3.119)

where kB is Boltzmann’s constant and T is the absolute temperature. We
describe the response of the atoms to the blackbody radiation in terms of the number of atoms na in the upper state and the number nb in the lower state.
Due to the three processes of spontaneous emission, stimulated emission, and
stimulated absorption, these numbers change according to the rate equations
n? a = ?Ana ? BU (?)(na ? nb) , (3.120)
n? b = +Ana + BU (?)(na ? nb) . (3.121)

We solve these equations in steady state, de?ned by n? a = n? b = 0. Either equation (note that n? a = ?n? b) gives
A/B
U (?) = n /n 1 ,


which with (3.118) becomes

b a ?

k?3/?2c3

U (?) = n /n

1 , (3.122)


Furthermore according to Boltzmann, in thermodynamic equilibrium the ra- tio of the number of atoms na in the upper state to that nb in the lower state
is given by
na = e?k?/kB T . (3.123)
nb
Substituting this into (3.120), we ?nd the Planck formula (3.119).


Rabi Flopping

Blackbody radiation is emitted by a collection of atoms in thermal equilib- rium with the radiation ?eld. On a microscopic basis the atoms constantly exchange energy with the ?eld in such a way that macroscopically no change is noticed. As we see in Sect. 5.1, this limit is valid in the rate equation approximation, for which the ?eld amplitude varies slowly (here not at all) compared to the atomic response.
Now let us consider the opposite extreme for which we ignore atomic damping altogether, and for simplicity we take the monochromatic ?eld
(3.113) with frequency ? approximately equal to the two-level transition fre- quency ? = ?a ? ?b. Examining the interaction energy (3.77, 3.81), we see that in the rotating-wave approximation we keep the e?i?t term for ?ni > 0. In the present case, ?> 0, and hence in the rotating-wave approximation we
keep only

1
Vab c ? 2 ?E0e?

i?t

. (3.124)

For Vba we use ei?t. Because ? may di?er somewhat from ?, it is convenient to write ?(r, t) slightly di?erently from (3.98), namely, as
. . 1 . .

?(r, t) = Ca(t)exp i

2 ? ? ?a

t ua(r)

+Cb(t)exp

. . 1 . .
i ? 2 ? ? ?b t

ub(r) , (3.125)

where the frequency shift ? = ? ? ?. This choice places the wave function in the rotating frame used for the Bloch vector in Sect. 4.3. Substituting this
expansion for ? into the Schro¨dinger equation (3.5) and projecting onto the eigenfunctions ua and ub as in the derivation of (3.20), we ?nd

1

C? a =
C? b =

2 i(??Ca + R0Cb) , (3.126)
1
2 i(?Cb + R?Ca) , (3.127)

where |R0| is the Rabi ?opping frequency de?ned by
?E0

R0 ?

(3.128)
k


after Rabi (1936), who studied the similar system of a spin– 1 magnetic dipole in nuclear magnetic resonance. Equations (3.126, 3.127) provide a simple ex- ample of two coupled equations, a combination we see repeatedly in phase conjugation (Chaps. 2, 10) and in linear stability analysis (Chap. 11). Equa-
tion (3.78) solves the n-level probability amplitude to ?rst order in the in-
teraction energy. Here, we solve the two-level probability amplitudes to all
orders in that energy.



Before solving (3.126, 3.127) generally, we can very quickly discover the basic physics by considering exact resonance, for which ? = 0. We can then
di?erentiate (3.127) with respect to time and substitute (3.128) to ?nd

1 2
¨ = ? 4 |R0| Cb ,

i.e., the di?erential equation for sines and cosines. In particular if at time
t = 0 the atom is in the lower state [Cb(0) = 1, Ca(0) = 0], then

1
Cb(t) = cos 2 |R0|t (3.129)


which from (3.127) gives




1
Ca(t) = i sin 2 |R0|t. (3.130)

The probability that the system is in the lower level |Cb(t)|2 = cos2 1 |R0|t =

(1 + cos|R0|t)/2, while |Ca|2 = sin2 1 |R0|t

= (1

? cos

|R0|t)/2. Hence the

wave function oscillates between the lower and the upper states sinusoidally at the frequency |R0|. In total contrast with blackbody radiation, instead of
coming to an equilibrium with constant probability for being in the upper
and lower levels, here the probabilities oscillate back and forth. In this case the atoms maintain a de?nite phase relationship with the inducing electric ?eld, while for blackbody radiation any such relationship averages to zero. As we see in Chap. 4 on the density matrix, Rabi ?opping preserves atomic coherence, while blackbody radiation destroys it. Further discussions on the irreversibility of coupling to a continuum are given in Sect. 14.3 on the theory of spontaneous emission, and more generally in Chap. 15.
To solve the coupled equations (3.126, 3.127) including a nonzero detuning
?, we write them as a single matrix equation
d . Ca(t) . = i . ?? R0 .. Ca(t) . . (3.131)

dt Cb(t)

? Cb(t)


This is a vector equation of the form dC/dt = 1 iMC, which has solutions of the form exp( 1 i?t). Accordingly substituting C(t) = C(0) exp( 1 i?t) into
2 2
(3.131), we ?nd that det(M ? ?I) = 0. This yields the eigenvalues ? = ±R,
where R is the generalized Rabi ?opping frequency
R? ,?2 + |R0|2 . (3.132)

Equation (3.131) has simple sinusoidal solutions of the form

1 1
Ca(t) = Ca(0) cos 2 Rt + A sin 2 Rt,


1 1
Cb(t) = Cb(0) cos 2 Rt + B sin 2 Rt.
Substituting these values into (3.126, 3.127) and setting t = 0, we immedi- ately ?nd the constants A and B. Collecting the results in matrix form, we
have the general undamped solution

. Ca(t) . . cos 1 Rt ? i?R?1

sinRt iR0R?1

sin 1 Rt

.. Ca(0) . .

Cb(t)

iR? ?1 1

1 ?1 1

C (0)

0 R sin 2 Rt cos 2 Rt + i?R

sin 2 Rt

b
(3.133)

The 2×2 matrix in this equation is precisely the Schro¨dinger evolution matrix
U (t) of (3.63) for the problem at hand. This U -matrix solution is valuable
for the discussion of coherent transients in Chap. 12 and in general whenever
damping can be neglected. It yields the ?rst-order perturbation result (3.115) in the limit of a weak ?eld (R ? ?). Section 4.1 shows how to account for
possibly unequal level decay from both levels. More general decay schemes
require the use of a density matrix as discussed in Sect. 4.2. Note that the matrix in (3.133) is a U matrix [see (3.62)]. For further discussion, see Probs.
3.14–3.16 and Sect. 14.1.