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of eqn (11.72) into eqn (11.71) leads to the integro-di¬erential equation

t
dC2 (t)
|gks | e’i∆k (t’t )
2
=’ dt C2 (t ) (11.75)
dt 0 ks

for C2 . This presents us with a di¬cult problem, since the evolution of C2 (t) now
depends on its past history. The way out is to argue that the function in curly brackets
decays rapidly as t ’ t increases, so that it is a good approximation to set C2 (t ) =
C2 (t). This allows us to replace eqn (11.75) by

t
dC2 (t)
|gks | e’i∆k t
2
=’ dt C2 (t) , (11.76)
dt 0 ks

which has the desirable feature that C2 (t + ∆t) only depends on C2 (t), rather than
C2 (t ) for all t < t. As we already noted in Section 9.2.1, evolutions with this property
are called Markov processes, and the transition from eqn (11.75) to eqn (11.76) is called
the Markov approximation. In the following paragraphs we will justify the assumptions
underlying the Markov approximation by a Laplace transform method that is also
useful in related problems.
¿¾ Coherent interaction of light with atoms

The di¬erential equations for C1 (t) and C2 (t) de¬ne a linear initial value problem
that can be solved by the Laplace transform method reviewed in Appendix A.5. Ap-
plying the general scheme in eqns (A.73)“(A.75) to the initial conditions (11.73) and
the di¬erential equations (11.71) and (11.72) produces the algebraic equations

ζ C2 (ζ) = 1 ’ gks C1ks (ζ) , (11.77)
ks


(ζ + i∆k ) C1ks (ζ) = g— C2 (ζ) . (11.78)
ks

Substituting the solution of the second of these equations into the ¬rst leads to
1
C2 (ζ) = , (11.79)
ζ + D (ζ)

where
2
|gks |
D (ζ) = . (11.80)
ζ + i∆k
ks

In order to carry out the limit V ’ ∞, we introduce

|gks |2 ,
g2 (k) = V (11.81)
s

which allows D (ζ) to be expressed as

g2 (k) g2 (k)
d3 k
1

D (ζ) = . (11.82)
(2π)3 ζ + i∆k
V ζ + i∆k
k

According to eqn (4.160),

ωk |K (∆k )|2 2
2
g (k) = |d| ’ d · k
2
, (11.83)
20

and the integral over the directions of k in eqn (11.82) can be carried out by the method
used in eqn (4.161). The relation |k| = ωk /c is then used to change the integration
variable from |k| to ∆ = ωk ’ ω21 . The lower limit of the ∆-integral is ∆ = ’ω21 ,
but the width of the cut-o¬ function is small compared to the transition frequency
ω21 ); therefore, there is negligible error in extending the integral to ∆ = ’∞
(wK
to get
3
2
|K (∆)|

∞ 1+
w21 ω21
D (ζ) = d∆ , (11.84)
2π ζ + i∆
’∞

where
2
|d| ω21
3
w21 = = A2’1 (11.85)
3π 0 c3
is the spontaneous decay rate previously found in Section 4.9.3.
¿¿
Spontaneous emission II


The time dependence of C2 (t) is determined by the location of the poles in C2 (ζ),
which are in turn determined by the roots of

ζ + D (ζ) = 0 . (11.86)

A peculiar feature of this approach is that it is absolutely essential to solve this equa-
tion without knowing the function D (ζ) exactly. The reason is that an exact evaluation
2
of D (ζ) would require an explicit model for |K (∆)| , but no physically meaningful
results can depend on the detailed behavior of the cut-o¬ function. What is needed is
an approximate evaluation of D (ζ) which is as insensitive as possible to the shape of
2
|K (∆)| . The key to this approximation is found by combining eqn (11.86) with eqn
(11.84) to conclude that the relevant values of ζ are small compared to the width of
the cut-o¬ function, i.e.
ζ = O (w21 ) wK . (11.87)
This is the step that will justify the Markov approximation (11.76). In the time do-
main, the function C2 (t) varies signi¬cantly over an interval of width ∆t ∼ 1/w21 ;
consequently, the condition (11.87) is equivalent to Tmem ∆t; that is, the memory
of the averaging function is short compared to the time scale on which the function
C2 (t) varies. The physical source of this feature is the continuous phase space of ¬nal
states available to the emitted photon. Summing over this continuum of ¬nal photon
states e¬ectively erases the memory of the atomic state that led to the emission of the
photon.
For values of ζ satisfying eqn (11.87), D (ζ) can be approximated by combining
the normalization condition K (0) = 1 with the identity

1 1
= πδ (∆) ’ iP ,
lim (11.88)
ζ’0 ζ + i∆ ∆

where P denotes the Cauchy principal value”see eqn (A.93). The result is
w21
D (ζ) = + iδω21 , (11.89)
2
where the imaginary part,
∞ 3 2
|K (∆)|
w21 ∆
=’
δω21 P d∆ 1 + , (11.90)
2π ω21 ∆
’∞

is the frequency shift. It is customary to compare δω21 to the Lamb shift (Cohen-
Tannoudji et al., 1992, Sec. II-E.1), but this is somewhat misleading. The result for
Re D (ζ) is robust, in the sense that it is independent of the details of the cut-o¬
function, but the result for Im D (ζ) is not robust, since it depends on the shape
2
of |K (∆)| . In Exercise 11.2, eqn (11.90) is used to get the estimate, δω21 /w21 =
O (wK /ω21 ) 1, for the size of the frequency shift. This is comforting, since it tells
us that δω21 is at least very small, even if its exact numerical value has no physical
signi¬cance. The experimental fact that measured shifts are small compared to the line
¿ Coherent interaction of light with atoms

widths is even more comforting. A strictly consistent application of the RWA neglects
all terms of the order wK /ω21 ; therefore, we will set δω21 = 0.
Substituting D (ζ) from eqn (11.89) into eqn (11.79) gives the simple result

1
C2 (ζ) = , (11.91)
ζ + w21 /2

and evaluating the inverse transform (A.72) by the rule (A.80) produces the corre-
sponding time-domain result
C2 (t) = e’w21 t/2 . (11.92)

Thus the nonperturbative Weisskopf“Wigner method displays an irreversible decay,

|C2 (t)|2 = e’w21 t , (11.93)

of the upper-level occupation probability. This conclusion depends crucially on the
coupling of the discrete atomic states to the broad distribution of electromagnetic
modes available in the in¬nite volume limit. In the time domain, we can say that
the atom forgets the emission event before there is time for reabsorption. We will see
later on that the irreversibility of the decay does not hold for atoms in a cavity with
dimensions comparable to a wavelength.
In addition to following the decay of the upper-level occupation probability, we can
study the probability that the atom emits a photon into the mode ks. According to
eqn (11.78),
g—
ks
C1ks (ζ) = . (11.94)
(ζ + i∆k ) (ζ + w21 /2)

The probability amplitude for a photon with wavevector k and polarization eks is
C1ks (t) ei∆k t , so another application of eqn (A.80) yields

e’i∆k t ’ e’w21 t/2
C1ks (t) = ig— . (11.95)
ks
∆k + iw21 /2

After many decay times (w21 t 1), the probability for emission is

2
pks = lim C1ks (t) ei∆k t
t’∞
2
|gks |
=
w21 2
2
(∆k ) + 2
2 2
ωk |K |d · eks |
(∆k )|
= . (11.96)
w21 2
2
20 V (∆k ) + 2


The denominator of the second factor e¬ectively constrains ∆k by |∆k | < w21 , so it is
permissible to set |K (∆k )| = 1 in the following calculations.
¿
Spontaneous emission II

As explained in Section 3.1.4, physically meaningful results are found by passing
to the limit of in¬nite quantization volume. In the present case, this is done by using
3
the rule 1/V ’ d3 k/ (2π) , which yields
2
|d · eks | d3 k
ωk
dps (k) = (11.97)
w21 2 (2π)3
2
20 (∆k ) + 2

for the probability of emitting a photon with polarization eks into the momentum-
space volume element d3 k. Summing over polarizations and integrating over the angles
of k, by the methods used in Section 4.9.3, gives the probability for emission of a photon
in the frequency interval (ω, ω + dω):
w21

2
dp (ω) = 2π. (11.98)
2
(ω ’ ω21 ) + w21
2

This has the form of the Lorentzian line shape
γ
L (ν) = , (11.99)
ν2 + γ2
where ν is the detuning from the resonance frequency, γ is the half-width-at-half-
maximum (HWHM), and the normalization condition is


L (ν) = 1 . (11.100)
π
’∞

From eqn (11.98) we see that the line width w21 is the full-width-at-half-maximum,
but also that the normalization condition is not exactly satis¬ed. The trouble is that
ω = ωk is required to be positive, so the integral over all physical frequencies is
∞ w21
dν 2
< 1. (11.101)
2
π ν 2 + w21
’ω21
2

This is not a serious problem since ω21 w21 , i.e. the optical transition frequency
is much larger than the line width. Thus the lower limit of the integral can be ex-
tended to ’∞ with small error. The spectrum of spontaneous emission is therefore
well represented by a Lorentzian line shape.

Two-photon cascade—
11.2.3
The photon indivisibility experiment of Grangier, Roget, and Aspect, discussed in
Section 1.4, used a two-photon cascade transition as the source of an entangled two-
photon state. The simplest model for this process is a three-level atom, as shown in
Fig. 11.1.
This concrete example will illustrate the use of the general techniques discussed
in the previous section. The one-photon detunings, ∆32,k = ck ’ ω32 and ∆21,k =
ck ’ ω21 , are related to the two-photon detuning, ∆31,kk = ck + ck ’ ω31 , by

∆31,kk = ∆32,k + ∆21,k = ∆32,k + ∆21,k . (11.102)
¿ Coherent interaction of light with atoms


!

kI




Fig. 11.1 Two-photon cascade emission from
kI
a three-level atom. The frequencies are as-
sumed to satisfy ω = ck ≈ ω32 , ω = ck ≈ ω21 ,

and ω32 ω21 .


According to the general result (11.43), the RWA Hamiltonian is

g32,ks e’i∆32,k t S32 aks + g21,ks e’i∆21,k t S21 aks + HC ,
Hrwa (t) = ’i (11.103)
ks

where the coupling constants are

ωk d32 · eks
g32,ks = K (∆32,k ) ,
2 0V
(11.104)
ωk d21 · eks
g21,ks = K (∆21,k ) .
2 0V

Initially the atom is in the uppermost excited state |µ3 and the ¬eld is in the
vacuum state |0 , so the combined system is described by the product state |µ3 ; 0 =
|µ3 |0 . The excited atom can decay to the intermediate state |µ2 with the emission of
a photon, and then subsequently emit a second photon while making the ¬nal transition
to the ground state |µ1 . It may seem natural to think that the 3 ’ 2 photon must
be emitted ¬rst and the 2 ’ 1 photon second, but the order could be reversed. The
reason is that we are not considering a sequence of completed spontaneous emissions,
each described by an Einstein A coe¬cient, but instead a coherent process in which
the atom emits two photons during the overall transition 3 ’ 1. Since the ¬nal states
are the same, the processes (3 ’ 2 followed by 2 ’ 1) and (2 ’ 1 followed by 3 ’ 2)
are indistinguishable. Feynman™s rules then tell us that the two amplitudes must be
coherently added before squaring to get the transition probability. If the level spacings
were nearly equal, both processes would be equally important. In the situation we
ω21 , the process (2 ’ 1 followed by 3 ’ 2) would be far o¬
are considering, ω32
resonance; therefore, we can safely neglect it. This approximation is formally justi¬ed
by the estimate
g32,ks g21,ks ≈ 0 , (11.105)
which is a consequence of the fact that the cut-o¬ functions |K (∆32,k )| and |K (∆21,k )|
do not overlap.
The states |µ2 ; 1ks = |µ2 |1ks and |µ1 ; 1ks , 1k s = |µ1 |1ks , 1k s will appear
as the state vector |˜ (t) evolves. It is straightforward to show that applying the
¿
Spontaneous emission II

Hamiltonian to each of these states results in a linear combination of the same three
states. The standard terminology for this situation is that the subspace spanned by
|µ3 ; 0 , |µ2 ; 1ks , and |µ1 ; 1ks , 1k s is invariant under the action of the Hamiltonian.
We have already met with a case like this in Section 11.2.2, and we can use the ideas
of the Weisskopf“Wigner model to analyze the present problem. To this end, we make
the following ansatz for the state vector:

|˜ (t) = Z (t) |µ3 ; 0 + Yks (t) ei∆32,k t |µ2 ; 1ks
ks

Xks,k s (t) ei∆31,kk t |µ1 ; 1ks , 1k s ,
+ (11.106)
ks k s


where the time-dependent exponentials have been introduced to cancel the time de-
pendence of Hrwa (t). Note that the coe¬cient Xks,k s is necessarily symmetric under
ks ” k s .
Substituting this expansion into the Schr¨dinger equation”see Exercise 11.5”
o
leads to a set of linear di¬erential equations for the coe¬cients. We will solve these
equations by the Laplace transform technique, just as in Section 11.2.2. The initial
conditions are Z (0) = 1 and Yks (0) = Xks,k s (0) = 0, so the di¬erential equations
are replaced by the algebraic equations

ζ Z (ζ) = 1 ’ g32,ks Yks (ζ) , (11.107)
ks



[ζ + i∆32,k ] Yks (ζ) = g—
32,ks Z (ζ) ’ 2 g21,k s Xks,k s (ζ) , (11.108)
ks


1—
Yk s (ζ) + g— s Yks (ζ) .
g
[ζ + i∆31,kk ] Xks,k s (ζ) = (11.109)
2 21,ks 21,k


Solving the ¬nal equation for Xks,k s and substituting the result into eqn (11.108)
produces

g21,k s g—
g—
21,ks
(ζ) ’
[ζ + i∆32,k + Dk (ζ)] Yks (ζ) = 32,ks Z Yk s (ζ) , (11.110)
ζ + i∆31,kk
ks


where
|g21,k s |2
Dk (ζ) = . (11.111)
ζ + i∆32,k + i∆21,k
ks

As far as the k-dependence is concerned, eqn (11.110) is an integral equation for

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