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where the ξa s are complex numbers. Substituting F into eqn (4.79) yields
n n

ξa ξb Fab 0, (4.83)
a=1 b=1

where F is the n — n hermitian matrix
Fab = G(1) (xa ; xb ) . (4.84)
Since the inequality (4.83) holds for all complex ξa s, the matrix F is positive de¬nite.
A necessary condition for this is that the determinant of F must be positive. For the
case n = 2 this yields the inequality
2
G(1) (x1 ; x2 ) G(1) (x1 ; x1 ) G(1) (x2 ; x2 ) . (4.85)

For ¬rst-order interference experiments, this inequality translates directly into a bound
on the visibility of the fringes; this feature will be exploited in Section 10.1.

4.8 The interaction picture
In typical applications, the interaction energy between the charged particles and the
radiation ¬eld is much smaller than the energies of individual photons. It is therefore
useful to rewrite the Schr¨dinger-picture Hamiltonian, eqn (4.29), as
o
(S) (S)
H (S) = H0 + Hint , (4.86)
where
(S) (S)
(S)
H0 = Hem + Hchg (4.87)
(S)
is the unperturbed Hamiltonian and Hint is the perturbation or interaction
Hamiltonian. In most cases the Schr¨dinger equation with the full Hamiltonian H (S)
o
½¾
The interaction picture

(S)
cannot be solved exactly, so the weak (perturbative) nature of Hint must be used to
get an approximate solution.
For this purpose, it is useful to separate the fast (high energy) evolution due to
(S) (S)
H0 from the slow (low energy) evolution due to Hint . To this end, the interaction-
picture state vector is de¬ned by the unitary transformation

Ψ(I) (t) = U0 (t) Ψ(S) (t) , (4.88)

where the unitary operator,
i (t ’ t0 ) (S)
U0 (t) = exp ’ H0 , (4.89)

satis¬es
‚ (S)
i
U0 (t) = H0 U0 (t) , U0 (t0 ) = 1 . (4.90)
‚t
Thus the Schr¨dinger and interaction pictures coincide at t = t0 . It is also clear
o
(S)
that H0 , U0 (t) = 0. A glance at the solution (3.76) for the Schr¨dinger equation
o
(S)
reveals that this transformation e¬ectively undoes the fast evolution due to H0 . By
contrast to the Heisenberg picture de¬ned in Section 3.2, the transformed ket vector
(S)
still depends on time due to the action of Hint . The consistency condition,

Ψ(I) (t) X (I) (t) ¦(I) (t) = Ψ(S) (t) X (S) ¦(S) (t) , (4.91)

requires the interaction-picture operators to be de¬ned by

X (I) (t) = U0 (t) X (S) U0 (t) . (4.92)
(S)
For H0 this yields the simple result

(I) (S) (S)
H0 (t) = U0 (t) H0 U0 (t) = H0 , (4.93)
(I) (S)
which shows that H0 (t) = H0 = H0 is independent of time.
The transformed state vector Ψ(I) (t) obeys the interaction-picture Schr¨dinger
o
equation
‚ †
(S) (S) (S)
Ψ(I) (t) = ’H0 Ψ(I) (t) + U0 (t) H0 + Hint Ψ(S) (t)
i
‚t

(S) (S) (S)
= ’H0 Ψ(I) (t) + U0 (t) H0 U0 (t) Ψ(I) (t)
+ Hint
(I)
= Hint (t) Ψ(I) (t) , (4.94)

which follows from operating on both sides of eqn (4.88) with i ‚/‚t and using eqns
(4.90)“(4.93). The formal solution is

Ψ(I) (t) = V (t) Ψ(I) (t0 ) , (4.95)
½¾ Interaction of light with matter

where the unitary operator V (t) satis¬es

‚ (I)
i V (t) = Hint (t) V (t) , with V (t0 ) = 1 . (4.96)
‚t
The initial condition V (t0 ) = 1 really should be V (t0 ) = IQED , where IQED is the
identity operator for HQED , but alert readers will su¬er no harm from this slight abuse
of notation.
By comparing eqn (4.92) to eqn (3.83), one sees immediately that the interaction-
picture operators obey
‚ (I)
X (t) = X (I) (t) , H0 .
i (4.97)
‚t
These are the Heisenberg equation for free ¬elds, so we can use eqns (3.95) and (3.96)
to get
aks (t) = aks e’iωk (t’t0 ) ,
(I) (S)
(4.98)
and
(S)
A(I)(+) (r, t) = aks eks ei[k·r’ωk (t’t0 )] . (4.99)
2 0 ωk V
ks

In the same way eqn (3.102) implies

F (I)(±) (r, t) , G(I)(±) (r , t ) = 0 , (4.100)

where F and G are any of the ¬eld components and (r, t), (r , t ) are any pair of
space“time points.
In the interaction picture, the burden of time evolution is shared between the oper-
ators and the states. The operators evolve according to the unperturbed Hamiltonian,
and the states evolve according to the interaction Hamiltonian. Once again, the density
operator is an exception. Applying the transformation in eqn (4.88) to the de¬nition
(3.85) of the Schr¨dinger-picture density operator leads to
o

‚ (I) (I)
ρ (t) = Hint (t), ρ(I) (t) ,
i (4.101)
‚t
so the density operator evolves according to the interaction Hamiltonian.
In applications of the interaction picture, we will simplify the notation by the fol-
lowing conventions: X (t) means X (I) (t), X means X (S) , |Ψ (t) means Ψ(I) (t) , and
ρ (t) means ρ(I) (t). If all three pictures are under consideration, it may be necessary
to reinstate the superscripts (S), (H), and (I).

4.8.1 Time-dependent perturbation theory
In order to make use of the weakness of the perturbation, we ¬rst turn eqn (4.96) into
an integral equation by integrating over the interval (t0 , t) to get
t
i
V (t) = 1 ’ dt1 Hint (t1 ) V (t1 ) . (4.102)
t0
½¾
The interaction picture

The formal perturbation series is obtained by repeated iterations of the integral equa-
tion,
2
t t t1
i i
V (t) = 1 ’ dt1 Hint (t1 ) + ’ dt2 Hint (t1 ) Hint (t2 ) + · · ·
dt1
t0 t0 t0

V (n) (t) ,
= (4.103)
n=0

where V (0) = 1, and
n t tn’1
i
’ dt1 · · · dtn Hint (t1 ) · · · Hint (tn ) ,
(n)
V (t) = (4.104)
t0 t0

for n 1.
If the system (charges plus radiation) is initially in the state |˜i then the prob-
ability amplitude that a measurement at time t leaves the system in the ¬nal state
|˜f is
Vf i (t) = ˜f |Ψ (t) = ˜f |V (t)| ˜i ; (4.105)
consequently, the transition probability is
2
Pf i (t) = |Vf i (t)| . (4.106)

4.8.2 First-order perturbation theory
For this application, we choose t0 = 0, and then let the interaction act for a ¬nite time
t. The initial state |˜i evolves into V (t) |˜i , and its projection on the ¬nal state
|˜f is ˜f |V (t)| ˜i . Let the initial and ¬nal states be eigenstates of the unperturbed
Hamiltonian H0 , with energies Ei and Ef respectively. According to eqn (4.104) the
¬rst-order contribution to ˜f |V (t)| ˜i is
t
i
(1)
Vf i (t) = ’ dt1 ˜f |Hint (t1 )| ˜i
0
t
i
=’ dt1 ˜f |Hint | ˜i exp (iνf i t1 ) , (4.107)
0

where we have used eqn (4.92) and introduced the notation νf i = (Ef ’ Ei ) / . Eval-
uating the integral in eqn (4.107) yields the amplitude

2i sin (νf i t/2)
(1)
Vf i (t) = ’ ˜f |Hint | ˜i ,
exp (iνf i t/2) (4.108)
νf i

so the transition probability is
2 4
(1) 2
| ˜f |Hint | ˜i | ∆ (νf i , t) ,
Pf i (t) = Vf i (t) = (4.109)
2


where ∆ (ν, t) ≡ sin2 (νt/2) /ν 2 .
½¾ Interaction of light with matter

2
For ¬xed t, the maximum value of |∆ (ν, t)| is t2 /4, and it occurs at ν = 0. The
width of the central peak is approximately 2π/t, so as t becomes large the function
is strongly peaked at ν = 0. In order to specify a well-de¬ned ¬nal energy, the width
must be small compared to |Ef ’ Ei | / ; therefore,


t (4.110)
|Ef ’ Ei |

de¬nes the limit of large times. This is a realization of the energy“time uncertainty
relation, t∆E ∼ (Bransden and Joachain, 1989, Sec. 2.5). With this understanding
of in¬nity, we can use the easily established mathematical result,

sin2 (νt/2)
∆ (ν, t) π
lim = lim = δ (ν) , (4.111)
2
t tν 2
t’∞ t’∞


to write the asymptotic (t ’ ∞) form of eqn (4.109) as

2π 2
t | ˜f |Hint | ˜i | δ (νf i )
Pf i (t) = 2
2π 2
t | ˜f |Hint | ˜i | δ (Ef ’ Ei ) .
= (4.112)

The transition rate, Wf i = dPf i (t) /dt, is then


| ˜f |Hint | ˜i |2 δ (Ef ’ Ei ) .
Wf i = (4.113)

This is Fermi™s golden rule of perturbation theory (Bransden and Joachain, 1989,
Sec. 9.3). This limiting form only makes sense when at least one of the energies Ei
and Ef varies continuously. In the following applications this happens automatically
because of the continuous variation of the photon energies.
In addition to the lower bound on t in eqn (4.110) there is an upper bound on the
time interval for which the perturbative result is valid. This is estimated by summing
eqn (4.112) over all ¬nal states to get the total transition probability Pi,tot (t) = tWi,tot ,
where the total transition rate is
2π 2
| ˜f |Hint | ˜i | δ (Ef ’ Ei ) .
Wi,tot = Wf i = (4.114)
f f


According to this result, the necessary condition Pi,tot (t) < 1 will be violated if
t > 1/Wi,tot . In fact, the validity of the perturbation series demands the more strin-
gent condition Pi,tot (t) 1, so the perturbative results can only be trusted for
1/Wi,tot . This upper bound on t means that the t ’ ∞ limit in eqn (4.111)
t
is simply the physical condition (4.110). For the same reason, the energy conserving
delta function in eqn (4.112) is really just a sharply-peaked function that imposes the
restriction |Ef ’ Ei | Ef .
½¾
The interaction picture

With this understanding in mind, a simpli¬ed version of the previous calculation is
possible. For this purpose, we choose t0 = ’T /2 and allow the state vector to evolve
until the time t = T /2. Then eqn (4.107) is replaced by
T /2
i
(1)
(T /2) = ’ ˜f |Hint | ˜i exp (iνf i T /2)
Vf i dt1 exp (iνf i t1 ) . (4.115)
’T /2

The standard result
T /2
dt1 eiνt1 = 2πδ (ν)
lim (4.116)
T ’∞ ’T /2

allows this to be recast as
2πi
(1) (1)
Vf i = Vf i (∞) = ’ ˜f |Hint | ˜i δ (νf i ) , (4.117)

so the transition probability is
2
2 2π
(1)
| ˜f |Hint | ˜i |2 [δ (νf i )]2 .
Pf i = Vf i = (4.118)

This is rather embarrassing, since the square of a delta function is not a respectable
mathematical object. Fortunately this is a physicist™s delta function, so we can use
eqn (4.116) once more to set
T /2
dt1 T
2
[δ (νf i )] = δ (νf i ) exp (iνf i t1 ) = δ (νf i ) . (4.119)
2π 2π
’T /2

After putting this into eqn (4.118), we recover eqn (4.113).

4.8.3 Second-order perturbation theory
Using the simpli¬ed scheme, presented in eqns (4.115)“(4.119), yields the second-order
contribution to ˜f |V (T /2)| ˜i :
2 T /2 t1
i
(2)
’ dt2 ˜f |Hint (t1 ) Hint (t2 )| ˜i
Vf i = dt1
’T /2 ’T /2
2 T /2 T /2
i
’ dt2 θ (t1 ’ t2 ) ˜f |Hint (t1 ) Hint (t2 )| ˜i , (4.120)
= dt1
’T /2 ’T /2

where θ (t1 ’ t2 ) is the step function discussed in Appendix A.7.1 . By introducing a
basis set {|Λu } of eigenstates of H0 , the matrix element can be written as

˜f |Hint (t1 ) Hint (t2 )| ˜i = exp [(iνf i ) T /2] ˜f |Hint | Λu Λu |Hint | ˜i
u
— exp (iνf u t1 ) exp (iνui t2 ) , (4.121)

where we have used eqn (4.92) and the identity νf u + νui = νf i . The ¬nal step is to
use the representation (A.88) for the step function and eqn (4.116) to ¬nd
½¿¼ Interaction of light with matter


˜f |Hint | Λu Λu |Hint | ˜i
i iνf i T /2
(2)
=’
Vf i e dν
2π 2 ν+i
’∞ u
— 2πδ (νf u ’ ν) 2πδ (νui + ν) . (4.122)

Carrying out the integration over ν with the aid of the delta functions leads to
˜f |Hint | Λu Λu |Hint | ˜i
2πi
(2)
Vf i = ’ δ (νf i )
2 νf u + i
u
˜f |Hint | Λu Λu |Hint | ˜i
= ’2πi δ (Ef ’ Ei ) . (4.123)
Ef ’ Eu + i
u

Finally, another use of the rule (4.119) yields the transition rate
2
˜f |Hint | Λu Λu |Hint | ˜i

δ (Ef ’ Ei ) .
Wf i = (4.124)
Ef ’ Eu + i
u


4.9 Interaction of light with atoms
4.9.1 The dipole approximation
The shortest wavelengths of interest for quantum optics are in the extreme ultraviolet,
so we can assume that » > 100 nm, whereas typical atoms have diameters a ≈ 0.1 nm.
The large disparity between atomic diameters and optical wavelengths (a/» < 0.001)
permits the use of the dipole approximation, and this in turn brings about important
simpli¬cations in the general Hamiltonian de¬ned by eqns (4.28)“(4.32).
The simpli¬ed Hamiltonian can be derived directly from the general form given in
Section 4.2.2 (Cohen-Tannoudji et al., 1989, Sec. IV.C), but it is simpler to obtain the
dipole-approximation Hamiltonian for a single atom by a separate appeal to the corre-
spondence principle. This single-atom construction is directly relevant for su¬ciently
dilute systems of atoms”e.g. tenuous atomic vapors”since the interaction between
atoms is weak. Experiments with vapors were the rule in the early days of quantum
optics, but in many modern applications”such as solid-state detectors and solid-state
lasers”the atoms are situated on a crystal lattice. This is a high density situation with
substantial interactions between atoms. Furthermore, the electronic wave functions can
be delocalized”e.g. in the conduction band of a semiconductor”so that the validity
of the dipole approximation is in doubt. These considerations”while very important
in practice”do not in fact require signi¬cant changes in the following discussion.
The interactions between atoms on a crystal lattice can be described in terms of
coupling to lattice vibrations (phonons), and the e¬ects of the periodic crystal potential
are represented by the use of Bloch or Wannier wave functions for the electrons (Kittel,
1985, Chap. 9). The wave functions for electrons in the valence band are localized to
crystal sites, so for transitions between the valence and conduction bands even the
dipole approximation can be retained. We will exploit this situation by explaining
the basic techniques of quantum optics in the simpler context of tenuous vapors. Once
these notions are mastered, their application to condensed matter physics can be found
elsewhere (Haug and Koch, 1990).

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