23

To see that this is the same as the position representation version of (3.2) is a little trickier. We substitute the identity expansion (3.34) in for the left bar
and the same expansion with r ? rr in for the right bar obtaining

¸
(O) =
¸
=

¸
d3r
¸
d3r


d3rr(?|r)(r|O|rr)(rr|?)

d3rr??(r, t)O(r, rr)?(rr, t) . (3.40)


Hermitian operators in quantum mechanics often turn out to be local, and hence
O(r, rr) = O(r, r)?(r ? rr) . (3.41)
This fact reduces (3.40) to (3.2) as desired. More generally we can interpret
O?(r, t) on the right-hand side of (3.2) as
¸

O?(r, t) =

d3rrO(r, rr)?(rr, t) .


Similarly substituting the expansion (3.37) into (3.39), we ?nd the ex- pectation value (3.24), where the operator matrix elements written earlier as (3.16) can be written simply as
Omn = (m|O|n) . (3.42)

Problem 3.2 shows that this equals (3.16). It is also useful to express the operator O directly in terms of the basis set. This then reads as
O = . Onm|n)(m| . (3.43)
n,m

Finally, we note that the state vector version of the Schro¨dinger equation (3.5) is




where

?
ik ?t |?) = H|?) , (3.44)

pˆ2
H = 2m + V (ˆr) . (3.45)

This reduces to the Schr¨odinger equation (3.5) in the coordinate representa- tion.


Coordinate and Momentum Representations

Equation (3.36) introduced the coordinate representation wave function
?(r, t) = (r|?). One can in a similar fashion introduce the momentum repre-
sentation wave function
?(p, t) = (p|?) . (3.46)


In order to discuss the connection between the coordinate and momentum representations, we restrict ourselves to a one-dimensional situation where
pˆ ? pˆ and xˆ ? xˆ, with [xˆ, pˆ] = ik. We proceed by introducing the translation,
or shift operator
S(?) = exp(?i?pˆ/k) . (3.47)
S(?) is unitary, with S†(?) = S?1(?) = S(??), and it has a number of interesting properties. For instance, using the commutation relations
[xˆ, f (pˆ)] = ikf r(pˆ) , (3.48)
[pˆ, g(xˆ)] = ?ikgr(xˆ) , (3.49)

which are proven in Prob. 3.19, one ?nds readily that
[xˆ, S(?)] = ?S(?) , (3.50)

so that xˆS(?) = S(?)[xˆ + ?]. Since the ket |x) is an eigenstate of the operator xˆ with xˆ|x) = x|x), we have therefore that xˆS(?)|x) = S(?)(x + ?)|x) = (x + ?)S(?)|x). In other words, the state S(?)|x) is also an eigenstate of xˆ, but with eigenvalue (x + ?): the action of the operator S(?) on the ket |x) is to transform it into a new eigenvector of xˆ with eigenvalue shifted by the arbitrary amount ?,
S(?)|x) = |x + ?) . (3.51)

This also proves that the spectrum of xˆ is continuous. Finally, we note that the coordinate representation wave function corresponding to the ket S(?)|?)
is
(x|S(?)|?) = (x ? ?|?) = ?(x ? ?) . (3.52)

The translation operator permits one to easily determine the action of the momentum operator pˆ in the coordinate representation. Considering a small shift ? such that S(??) = exp(i?pˆ/k) c 1 + i ? pˆ + O(?2), we have





so that

?
(x|S(??)|?) = ?(x) + i


(x|pˆ|?) + O(?2) = ?(x + ?) , (3.53)

(x|pˆ|?) =

k lim i ??0

?(x+ ?) ? ?(x)
?

= k d
i dx

?(x) . (3.54)

The action of pˆ in the coordinate representation is therefore

d
pˆ ? ?ik dx . (3.55)

A similar derivation shows that in the momentum representation, the action of xˆ is
d
xˆ ? ik dp . (3.56)


Armed with this knowledge, it is quite easy to obtain the form of the Schr¨odinger equation in the momentum representation from its coordinate
representation of (3.6). This requires evaluating (p|V (xˆ)|?(t)). Introducing the identity ¸ dp |p)(p| = 1, we have

¸
(p|V (xˆ)|?(t)) =

¸
dpr(p|V (xˆ)|pr)(pr|?(t)) =


dprV (p ? pr)?(pr, t) , (3.57)


where V (p ? pr) are of course the matrix elements of the potential V in
the momentum representation. They can be obtained from the coordinate
representation by noting that

¸
(p|V (x)|pr) =


dx(p|x)(x|V (xˆ)|x)(x|pr) . (3.58)


Since (x|pr) may be interpreted as the coordinate representation wave func- tion ?(x) associated with the state vector |pr) we have from (3.54)
d?(x)



so that

(x|pˆ|p) = ?ik

dx ,

?(x) =

. ? ik . d?(x)

p dx
or
1
(x|p) = ? exp(ipx/k) , (3.59) 2?k
where we have used the plane wave normalization appropriate for the one- dimensional situation at hand. Introducing this result into (3.58) yields

1 ¸
V (p) = ?
2?k


dx e


?ipx/k


V (x) , (3.60)

that is, V (p) is the Fourier transform of V (x). With this result, the momen- tum representation of the Schro¨dinger equation is therefore

ik d?(p) =
dt

. p2 .
2m

1 ¸
?(p) +
(2?k)n/2


dnp V (p ? p )?(p ) (3.61)

r r r
Here, we have extended the one-dimensional result to n dimensions in a straightforward way. Note that in contrast to the coordinate representation, where the potential energy term is usually local and the kinetic energy term is not, the kinetic energy term is now local, but the potential energy term is not.


Schr¨odinger, Heisenberg and Interaction Pictures

The problem we are normally interested in quantum optics is to determine the expectation values (3.39) of observables at the time t. To do this we typ-
ically start with a system in a well de?ned state at an earlier time and follow the development up to the time t using the Schro¨dinger equation (3.44). It is
possible to follow this evolution in three general ways and in many combina-
tions thereof. The one we use primarily in the ?rst part of this book is called the Schro¨dinger picture and puts all of the time dependence in the state vec- tor. The interaction picture puts only the interaction-energy time dependence into the state vector, putting the unperturbed energy dependence into the operators. The Heisenberg picture puts all of the time dependence into the operators, leaving the state vector stationary in time. In the remainder of this section, we review the way in which these three pictures are tied together.
The Schr¨odinger equation (3.44) can be formally integrated to give
|?(t)) = U (t)|?(0)) , (3.62)
where the evolution operator U (t) for a time-independent Hamiltonian is given by
U (t) = exp(?iHt/k) . (3.63)
Substituting (3.62) into the expectation value (3.39) we obtain
(O) = (?(0)|U †(t)O(0)U (t)|?(0)) . (3.64)

We can also ?nd this same value if we can determine the time dependent operator.
O(t) = U †(t)O(0)U (t) . (3.65)
As Heisenberg ?rst showed, it is possible to follow the time evolution of the quantum mechanical operators. In fact we can obtain their equations of motion by di?erentiating (3.65). This gives


d dU †

?O dU



i.e.,

dt O(t) =

dt OU + U † ?t U + U †O dt ,

d i

?O

dt O(t) = k [H, O] + U

U , (3.66)
?t

where

[H, O] ? HO ? OH (3.67)

is the commutator of H with O. In deriving (3.66), we have used the fact that H commutes with U , which follows from the fact that U is a function of H only. The ?O/?t accounts for any explicit time dependence of the Schro¨dinger operator O.


In general when the system evolution is determined by integrating equa- tions of motion for the observable operators, we say the Heisenberg picture is being used. When the evolution is determined by integrating the Schro¨dinger equation, we say that the Schro¨dinger picture is being used. In either case, (3.64) shows that we get the same answers. You ask, why use one picture instead of the other? The answer is simply, use the picture that makes your life easier. Typically the insights obtained with one di?er somewhat from the other, but you get the same answer with either. Traditionally the Schro¨dinger picture is the ?rst one taught to students and many people feel more com- fortable with it. Much of this book is carried out in the Schro¨dinger picture.
On the other hand, the Heisenberg picture is a “natural” picture in the sense that the observables (electric ?elds, dipole moment, etc.) are time-
dependent, exactly as in classical physics. As a result, their equations of motion usually have the same form as in the classical case, although they are operator equations, which modi?es the way one can integrate and use them.
Another aspect is that in the Schro¨dinger picture, one has to ?nd |?(t))(or its generalization the density operator ?) before computing the desired expec- tation values. Since |?(t)) contains all possible knowledge about the system,
you have to solve the complete problem, which may be more than you need.
In many cases, you only want to know one or a few observables of the system. The Heisenberg picture allows you to concentrate on precisely those observ- ables, and with some luck, you may not have to solve the whole problem to get the desired answers.
In discussing (3.13, 3.22), we hinted at another way of following the time
dependence, namely, we put only the time dependence due to the interaction energy into the Cn(t), while the time dependence of the total Hamiltonian is contained in the cn(t). The state vector of (3.37) is the Schr¨odinger-picture
state vector, while the state vector
|?I (t)) = . Cn(t)|n) (3.68)
n

is said to be the interaction-picture state vector. The thought behind using the interaction picture is to take advantage of the fact that we often face situations where we already know the solutions of the problem in the absence of the interaction.
More formally, to eliminate the known part of the problem, we substitute the state vector



where

|?S (t)) = U0(t)|?I (t)) , (3.69)

U0(t) = exp(?iH0t/k) (3.70)

into the Schro¨dinger equation (3.44). We include the subscript S in (3.69) to remind ourselves that |?S(t)) is the Schr¨odinger-picture state vector. We
?nd

d i
dt |?I (t)) = ? k VI (t)|?I (t)) , (3.71)
where we have de?ned the interaction-picture interaction energy
VI (t) = U †(t)VS U0(t) (3.72)
and put a subscript S on the RHS to remind ourselves that VS is in the Schr¨odinger picture. From (3.69, 3.39), we also immediately ?nd that the
ex-ceptation value of an operator O in the interaction picture is given by
(O(t)) = (?I (t)|OI (t)|?I (t)) , (3.73)


where


OI (t) = U †(t)OS U (t)0 . (3.74)

Note that since we know the solution of the unperturbed problem, OI (t) is
already known. Comparing the equation of motion (3.71) achieved with the
original Schro¨dinger equation (3.44), we see that we have achieved our goal, namely, that we have eliminated the part of the problem whose solution we al- ready knew. We see in Chap. 4 that the interaction picture (or more precisely, an interaction picture) is particularly helpful in visualizing the response of a two-level atom to light.


3.2 Time-Dependent Perturbation Theory

To predict expectation values of operators, we need to know what the wave function is. Typically, we know the initial value of the wave function, which then evolves in time according to the Schro¨dinger equation, or equivalently, the operators of interest evolve according to the Heisenberg equations. For some problems, these equations can be integrated exactly, giving us the values
needed to compute the expectations values and the desired time t. More gen-
erally the equations can be integrated approximately using a method called
time-dependent perturbation theory. The name comes from the introduction of a perturbation energy V as given in (3.18), which describes the interaction
of the quantum system under consideration with some other system. An atom
interacting with an electromagnetic ?eld is the combination that we consider most often in this book. The perturbation energy forces the probability am- plitudes in (3.13) or (3.22) to be time dependent. The method of time de- pendent perturbation theory consists of formally integrating the Schro¨dinger equation, converting it into an integral equation, and then solving the in-
tegral equation iteratively. One way to proceed is by writing V ? ?V and
|?(t))? |?(0)(t)) + ?|?(1)(t)) + ?2|?(2)(t)) + .. .. The zeroth-order solution,
the solution in absence of perturbation, is then obtained by equating terms proportional to order ?0 on both sides of Schr¨odinger’s equation. First-order perturbation theory is obtained by equating terms proportional to ?, second- order theory to terms proportional to ?2, and so on. One can then set ? = 1



at the end of the calculation. For example on the RHS of the equations of motion (3.20) for the Cn, we insert the initial values of the Cn and inte-
grate, obtaining better values on the LHS. This ?rst integration gives the “?rst-order” corrections to the Cn. That may be accurate enough for your
purposes, and it is used in the famous Fermi Golden Rule. If it is not accurate
enough, you substitute the improved values in on the RHS and integrate to obtain a second-order correction. One can iterate this procedure to succes- sively higher orders of perturbation. This section carries out this procedure to ?rst order in the perturbation energy, i.e., one time integration. The answer is illustrated and then used to derive the Fermi Golden Rule. The section concludes with a general formulation of higher-order perturbation theory.
An important question is, given a quantum system initially in the state
|i), what are the probabilities that transitions occur to other states? This
question asks, for example, what the probability is that an initially unexcited
atom interacting with an electromagnetic ?eld absorbs energy from the ?eld.
The wave function (3.13) has the initial value
?(r, 0) = ui(r) ,

that is,


Ci(0) = 1, Cnƒ=i(0) = 0 . (3.75)

To ?nd out the ?rst-order correction to the Cn(t), we use the initial values (3.75) on the RHS of the Schro¨dinger equations of motion (3.20) for the Cn(t).
This gives
C? n(t) c C? (1)(t) = ?ik?1(n|V|i)ei?ni t , (3.76)
where C(1) is a special case of C(k) , which means we have iterated (3.20) k
n n
times.
Equation (3.76) is easy to integrate for two important kinds of perturba- tion energies: one time independent, and one sinusoidal such that
V = V0 cos ?t . (3.77)
Integrating (3.76) from 0 to t for a time independent V(? = 0), we have
ei?ni t ? 1

Cn(t) c C(1)(t) = ?ik?1Vni

i?ni


sin (?ni t/2)

= ?ik?1Vniei?ni t/2

,
?ni/2

where we write Vni for (n|V0|i). The probability that a transition occurs to level n is given by

(1) 2
| n |

= |Vni

|2 sin2(?ni

t/2)

. (3.78)

k2 (?ni/2)2
It’s interesting to note that we have already seen this kind of result in the phase matching discussion of Sect. 2.2, for which electromagnetic ?eld am- plitudes are used instead of the probability amplitudes used here. Problem



3.3 discusses the analogy between the two problems. The value of (3.78) is accurate so long as Ci(t) doesn’t change appreciably from the initial value Ci(0) = 1. In view of the normalization condition (3.14), this means that the
total transition probability
PT = 1 ? |C(1)|2 = . |C(1)|2 (3.79)

i


must be much less than unity.

n nƒ=i

Figure 3.2 plots the probability in (3.78) at the time t as a function of the frequency di?erence ?ni. For short enough times we can expand the sine
in (3.78) to ?nd

|C(1) 2

2
|Vni| 2

n | c

k2 t

(3.80)

which shows that the center of the curve increases proportionally to t2. We further see that for increasing frequency di?erences |?ni|, the probability that the interaction induces a transition to level n becomes smaller rapidly. Thus
transitions are much more likely if the energy is conserved between initial
and ?nal states.
Consider now the sinusoidal interaction energy (3.77), which can be used to model an atom interacting with a monochromatic electromagnetic ?eld. For such a ?eld, (3.77) is proportional to the electric ?eld amplitude, as we see in Sect. 3.3. Integrating (3.76) accordingly, we have

. ei(?ni +? )t 1
Cn(t) c C(1)(t) = ?i

ei(?ni +?)t 1 .
+


. (3.81)

n 2k

i(?ni + ?)

i(?ni ? ?)


For the sake of de?niteness, consider the case ?ni > 0. Then the denomina- tor ?ni + ? is always positive and larger than ?ni. This is not true for the


.5




.25




0
0 2? 3? 4?
(? ? ?) t

Fig. 3.2. Probability |Cn (t)|2 of (3.78) versus (? ? ?)t/2


denominator ?ni ? ?, which vanishes if the resonance condition
? c ?ni (3.82)

is satis?ed. For interactions near resonance, the term with the relatively small denominator ?ni ? ? is much larger than that with the ?ni + ?, allowing us to
neglect the latter. For the same reason, we can probably neglect transitions to levels with energies very di?erent from k?. This observation is used to
justify the two-level atom approximation discussed in Sect. 3.3. Neglecting the term with the relatively large denominator ?ni + ? is called the rotating-
wave approximation. It is used in much of the book.
Making the rotating-wave approximation in (3.81), we ?nd the transition probability


(1) 2
| n |

= |Vni

|2 sin2[(?ni

? ?)t/2]

. (3.83)

4k2

(?ni ? ?)2/4


This result is formally the same as the dc case of (3.78), provided we substi- tute ?ni ? ? for ?ni. Thus Fig. 3.2 and the corresponding discussion apply to
this case as well. In particular, we see that in the course of time, transitions
are unlikely to occur unless the resonance condition (3.82) is satis?ed, that is, unless the applied ?eld frequency matches the transition frequency.
So far we have in?nitely sharp energy levels. This is not realistic, since levels can be broadened by e?ects like spontaneous emission and collisions. Furthermore, there may be a continuum of levels such as in the energy bands in solid-state media. For these situations, the summation in the total transi- tion probability (3.79) can be replaced by an integral with a density of state
factor D(?) to weight the distribution correctly. For example, there are typi-
cally more states per frequency interval for higher frequencies then for lower frequencies. The total transition probability PT then has the value
¸

PT c

D(?)|C(1)(?)|2d?,


where the discrete frequency ?ni is replaced by the continuous frequency ?.
Substituting (3.83), we ?nd


¸
PT =

d? D(?) |V

(?)|2

sin2[(? ? ?)t/2]


. (3.84)

4k2

[(? ? ?)t/2]2


It is interesting to evaluate the total transition probability integral (3.84) for two reasons. First, so long as it is small enough, we know that the ?rst-
order perturbation theory answer is valid. Secondly, dPT /dt gives the rate at
which transitions occur. The equation for this rate is called the Fermi Golden
Rule, and can be used to ?nd a variety of rates, such as those occurring in the photoelectric e?ect, spontaneous emission, and the Planck radiation law.
Equation (3.84) is a special case of the general integral

¸
J = d?F (?)G(?) . (3.85)

There are problems for which both the density of states factor D(?) and the matrix elements Vni are known. But even in such situations the resulting in- tegral for (3.84) is typically hard to solve. However, we can approximate J if either F (?) or G(?) varies little over the frequency range for which the other
has an appreciable value. The extreme example is when one of the functions, say G(?) is the delta function ?(? ? ?0). Then J = F (?0). More generally, suppose G(?) is sharply peaked about ?0 and that F (?0) varies little in this interval. Then J c F (?0) ¸ G(?). For the purposes of this problem, G(?) is
a delta function, and it is in this way that delta functions approximate nat-
ural behavior. This is an example of what one sometimes calls an “adiabatic elimination”. This kind of elimination is equally important in the solution of coupled di?erential equations, for which one function varies slowly compared to another. Typically we consider problems in which atoms coupled to an electromagnetic ?eld vary rapidly compared to the ?eld envelope. In such cases, the technique of adiabatic elimination allows us to solve the atomic equations of motion assuming that the ?eld envelope is constant, and then to substitute the resulting steady-state polarization of the medium into the correspondingly simpli?ed slowly-varying ?eld equations of motion (Chap. 5).
2

The integral (3.84) has the form of (3.85) with F (?) = D(?)|V(?)|

and

G(?) = sin2[(???)t/2]/[(w??)t/2]2. Hence we can solve (3.84) when either F or G varies rapidly compared to the other. In particular, for times su?ciently small that all relevant values of |? ? ?|t are much less than unity, the G = sin x/x function in (3.84) can be approximated by unity. By relevant values of |? ? ?|t, we mean those for which the density of states factor D(?) and the matrix element V(?) have appreciable values. This then gives
¸ D(?)|V(?)|2

PT c t2


4k2

d?. (3.86)

Hence for such small times, the transition rate dPT /dt is proportional to time, starting up from zero. Unless the density of states and the interaction energy matrix element have in?nitely wide frequency response, i.e., in?nite bandwidth, this limit implies a build-up time in the response of the system
to the applied perturbation V(?). (This means, for example, that detectors
have a nonzero response time).
After this initial small time region, we suppose that the factor D(?)|Vni|2 varies little in the frequency interval for which the sin2 x/x2 function in (3.84) has appreciable values (see Fig. 3.3). In this limit, we can evaluate D(?)|Vni|2 at the peak ? = ? of the sin2 x/x2 function, ?nding

PT = D(?) |V

(?) 2 ¸ ?
t


sin2x dx

= ?

2 2 x2
??
2

2k2 D(?)|V(?)| t. (3.87)


1.00


0.75



sin2x x2


0.50



0.25



0.00
?2? ?? 0
x



? 2?

Fig. 3.3. sin2 x/x2 term in the total transition probability integral (3.87) versus
x = (? ??)t/2. As time increases, this function peaks up like the ?-function ?(? ??)


Here, we have extended the limits of the integral to ±? since in the present approximation this adds little to the integral and yields an analytic answer. Equation (3.87) gives then the Fermi Golden Rule rate


dPT

d (1) 2 ? 2

? = = C
dt dt

= D(?)|V(?)|

, (3.88)

which is a constant in time.
This constant rate proportional to the intensity of the incident radiation is what people typically observe in the photoelectric e?ect. Note that the
rate vanishes if no transitions exist for the frequency ?. The photoelectric
e?ect occurs in media that have an energy gap above the ground state. To be absorbed, the applied photon energy k? must be larger than this gap.
Summarizing the rate at which transitions occurs, we see that for times short compared to the reciprocal of the width of the function |V0|2D, the rate
increases linearly in time. For longer times the rate becomes constant. For still longer times, when PT does not remain small, we cannot assume that
the probability of the initial state is unity. Section 14.3 shows that to a good
approximation, we can ?x up the rate by multiplying it by the initial state probability |Ci(t)|2. This generalizes the Fermi Golden Rule to


d 2
dt |Ci|


= ?? |Ci|2


, (3.89)

which states that the probability for being in the initial level decays exponen- tially in time. Such a formula doesn’t make a ?rst-order perturbation theory


approximation. This kind of time response is typical of certain important pro- cesses in quantum optics, such as spontaneous emission, which is described in Chap. 14.
Note that we get the same kind of integral (3.84) for two sharp levels
interacting with a non-monochromatic ?eld with a spectral intensity distri- bution proportional to V(?). This yields a transition rate ? given by (3.88) with ? replaced by ?, since it is the atomic frequency ?, rather than the ?eld frequency ?, that determines the center of the sin2 x/x2 distribution. This
observation is important in the next section, where we derive the Planck
black-body radiation formula.


Higher-Order Perturbation Theory

The iterative approach outlined at the start of this section can be written in an analytic form by using the formal solution of (3.62) in the interaction picture as
|?I (t)) = UI (t)|?I (0)) . (3.90)
Here, we use the subscript I to remind ourselves that we are working in the interaction picture. Taking the time rate of change of (3.90) and using the Schr¨odinger equation (3.71), we obtain
dUI (t)
ik = VI (t)UI (t) . (3.91)

Remembering that UI (0) = 1, we integrate this equation formally to get
i ¸ t

UI (t) = 1 ? k

dt VI (t )UI (t ) . (3.92)

r r r
0

We can solve this equation by successive iterations, obtaining
i ¸ t

UI (t) = 1 ? k

dt1 VI (t1)
0

. i .2 ¸ t
+

dt1 VI (t1)

¸ t1

dt2 VI (t2) + ... . (3.93)

k 0 0
Truncating this expression after the lowest order term in VI gives ?rst-order perturbation theory. Keeping higher-order terms gives second-order, third- order, etc., perturbation theory. Note that this iteration process implies a
time ordering such that t2 ? t1 ? t.
By way of illustration, we calculate the ?rst-order answer this way as

.
|?I (t))c

i ¸ t
1 ?
0

.
dt1 VI (t1)


|?I (0)) . (3.94)

Substituting this into the equation for the transition probability to level m






we have

|Cm(t)|2 = |(m|?I (t))|2 = |(m|?S (t))|2 , (3.95)

¸

|Cm(t)|2 c |(m|i)|2 + k?2|(m|

dt1VI (t1)|i)|2 .


Converting VI back to the Schro¨dinger value using (3.72), we have

2
1 .¸ t r

|Cm(t)|2 c |C(1)(t)|2 =

.
k2 . 0

dt ei?mi t (m|VS |i).
.

. (3.96)


This gives (3.83) as before. By including more terms in (3.94), we can calcu- late successively higher-order contributions.


3.3 Atom-Field Interaction for Two-Level Atoms

This section introduces the two-level atom, a concept we write a great deal about in this book. Such later consideration merits a careful introduction. Consider ?rst the simplest of all atoms, hydrogen. This atom has an in?nite number of bound levels, characterized by the energies
e R ?
En = ? 2a n2 = ? n2 , (3.97)
where n = 1, 2, 3,... , a0 is the Bohr radius (a0 = 0.53 ?A), and R? = 13.6 eV is Rydberg’s constant. A few of these energy levels are shown in Fig. 3.4.


0.00


?0.25

?0.50
En
?0.75

?1.00


0 2 4 6 8
r

Fig. 3.4. Energy levels of the hydrogen atom units of R?


Unlike the quantum simple harmonic oscillator of Sect. 3.4, the energy levels of hydrogen and of atoms in general are not equally spaced. For example,

3 5
E2 ? E1 = 4 R? , E3 ? E2 = 36 R? .
In quantum optics and in laser spectroscopy, we often shine monochro- matic laser light of such an atom and study what happens. If the laser fre- quency almost matches a particular transition frequency, then the transition probability predicted by (3.83) for this transition is much larger than that for other transitions. The approximation is almost the same as that used in making the rotating-wave approximation: in both cases, one neglects terms with denominators large compared to the term with the resonant denomina-
tor. A particular frequency di?erence ?ni ? ? is much smaller, say 2?×108 radians/sec, than the sum ?ni + ?(c 2?×1014 radians/sec) for the antiro- tating wave or the di?erence ?mi ? ?, m ƒ= n, which might be 0.6?×1014
radians/sec for some nonresonant transition. If this is the case, the problem
reduces to two levels. Since the antirotating-wave contribution is actually smaller than many nonresonant contributions, it follows that the two-level atom approximation is usually only consistent if made simultaneously with the rotating-wave approximation. If one decides to keep the antirotating wave contribution, one must also keep all the nonresonant contributions as well. This is not as hard as it might seem, since nonresonant contributions can be usually treated using ?rst-order perturbation theory. Also, we can account to some degree for transitions to levels other than the principal two by including various decay and pump rates.
A famous two-level system is the spin 1/2 magnetic dipole in nuclear
magnetic resonance. This is a true two-level system with relatively simple decay mechanisms. It has a lot in common with its brother the two-level atom, but its response can di?er signi?cantly in cases where level decay rates play an important role.
In our treatment of atoms using the two-level approximation, we ignore the fact that levels usually have a number of sublevels that all can contribute
to a resonant transition. This produces complications when experiments with real atoms are used to test theories based on the two-level approximation. In such cases, optical pumping techniques can sometimes be used to produce a true two-level atom.
We emphasize the two-level atom because we can often describe its inter-
action with the electromagnetic ?eld in detail and obtain analytic solutions. It thus allows us to learn a great deal about the atom-?eld interaction, and hopefully this knowledge can be generalized to more realistic situations. Note
that although the two-level atom includes the low-order ?(3) type of nonlin-
earity of Sect. 2.3 as a special case, in general it provides for more complicated
nonlinear responses, such as saturation.
We label the upper level of our two-level atom by the letter a, and the lower by b as shown in Fig. 3.5. The corresponding wave function is














Fig. 3.5. Energy level diagram of two-level atom

?(r, t) = Ca(t)e?i?a tua(r) + Cb(t)e?i?b tub(r) . (3.98) Before we see how this wave function evolves under the in?uence of an
applied electromagnetic ?eld, let us consider the kind of charge distribution it represents. To be speci?c, suppose the lower level is the 1s ground state of
hydrogen with the energy eigenfunction
ub(r) = u100(r, ?, ?) = (?a3)?1/2e?r/a0 , (3.99) and the upper level is the 2p state with the eigenfunction
ua(r) = u210(r, ?, ?) = (32?a3)?1/2(r/a0) cos ? e?r/2a0 . (3.100) These eigenfunctions are plotted versus the z coordinate in Fig. 3.6, followed
by the superposition ?(r, t) of (3.98) for two times separated by ?/?. For one
of these times, the two probability amplitudes in (3.98) add. Half a period
2

later, they subtract. Figure 3.6c shows the probability density |?(z, t)|

for

these two points in time. We see that this probability density, and hence the “charge density” e|?(z, t)|2, oscillates back and forth across the nucleus in
a fashion analogous to the charge on the spring in Sect. 1.3. This similarity
is the underlying reason why the classical model of Chap. 2 is so successful in describing the linear absorption of light by a collection of atoms. Chapter 5 derives this response quantum mechanically in detail, revealing where the classical model fails in laser physics.


Electric dipole interaction

We mentioned in the discussion of the free-electron laser of Chap.1 that the interaction between light and charged particles is described by invoking the principle of minimum coupling , which states that the canonical momentum
p of a particle of charge q is no longer its kinetic momentum mr? , as is the
case for a free particle, but rather
p = mr? + qA(r) , (3.101)


where A(r) is the vector potential and U (r) the scalar potential and
E = ?A U ,
?t
B = ?× A . (3.102)

As discussed at length in Problems (3.20) – (3.212) the classical version of this Hamiltonian guarantees that charged particles are subjected to the Lorentz force, as should be the case, and also that the electromagnetic ?eld is governed by Maxwell equations.1 More formally, it follows from the requirement of local gauge invariance, which states that the physical predictions of our theory must remain unchanged under the gauge transformation
?(r, t) ? ?(r, t)ei?(r,t) .

The interaction between a light ?eld a charge q bound to an atomic nucleus by a potential V (r) is then given in the non-relativistic limit by the Hamiltonian


1 2
H = 2m [p ? qA(r, t)]


+ qU (r, t) + V (r) , (3.103)

where the vector potential A and the scalar potential U are evaluated at the location r of the charge. We recognize that the ?rst term in that Hamiltonian
is just the kinetic energy of the charged particle.
We are free to choose to work in the so-called radiation gauge, where
U (r, t) = 0 (3.104)


and


?• A(r, t) = 0 , (3.105)

and we do so consistently in the remainder of this book. In addition, we exploit the fact that in most problems of interest in quantum optics, the wavelength of the optical ?eld is large compared to the size of an atom, and it is justi?ed to evaluate the vector potential at the location R of the nucleus rather than at the location r of the electron. This amounts to approximating that ?eld as constant over the dimensions of the atom, and is called the electric dipole approximation, or dipole approximation in short.
With the coordinate representation form of the canonical momentum p =
?ik?, the Schro¨dinger equation becomes then

??(r,t)

k2 . iq .2

ik =
?t 2m

?? k A(R, t)

+ V (r) . (3.106)

Introducing the new wave function ?(r, t) via the gauge transformation

1 Theoretically inclined students are strongly encouraged to work through these problems in detail.


?(r, t) = exp[(?iqr/k) • A(R, t)]?(r, t) , (3.107)
and remembering that [p, f (x)] = ?ikf r(x) and that in the Coulomb gauge the electric ?eld and the potential vector are related by
?A(R, t)

E(R, t) = ?

(3.108)
?t

we ?nd that ?(r, t) obeys the Schro¨dinger equation
?(r, t) = [H0 ? qr • E(R, t)]?(r, t) . (3.109)
where H0 = p2/2m + V (r) is the unperturbed Hamiltonian of the electron. This shows that in the electric dipole approximation, the interaction between the electron and the electromagnetic ?eld is described by the interaction Hamiltonian
V = ?qr • E(R, t) , (3.110)
where R is the position of the center of the mass of the atom.2
Typically we are also interested in plane waves, for which we write simply
E(z, t), where z is the axis of propagation. The dipole traditionally is writ-
ten as the positive charge value times the distance vector pointing from the negative to the positive charge. This gives the same answer as er, which is
the negative charge value of the electron times the distance vector pointing
from the positive charge to the negative charge. There has been substantial discussion over the years since Lamb (1952) ?rst brought it up concerning
the use of (3.110) versus a Hamiltonian involving A • p, where A is the ?eld
vector potential and p is the electron momentum. For our purposes, (3.101)
combines intuitive appeal with excellent accuracy.
The matrix element of the dipole operator between a level and itself [recall (3.16)]

¸
er?? =

d3r er|u?(r)|2 , (3.111)


vanishes unless the system has as permanent dipole moment (like H2O), since
|u?(r)|2 is inevitably a symmetrical function of r and r itself is antisymmetric.
Matrix elements of r between di?erent states can also vanish, but we are primarily interested in two levels a and b between which the matrix element
does not vanish. We can then write the electric-dipole interaction energy
matrix element
Vab = ??E(R, t) , (3.112)

where ? (pronounced “squiggle” and also used for the Weierstrass elliptic function) is the component of erab along E.

2 For an excellent discussion of the Hamiltonian approach to electrodynamics, the electric dipole interaction, and the the A • p vs. E • r forms of the electric dipole interaction, see Cohen-Tannoudji, Dupont-Roc and Grynberg (1989).






(a)








(b)

u100 (z)








u100 (z)

+ u210 (z)

z +




? u210 (z)

z ?

= ?(z, t1)

z = z




= ?(z, t2)

z = z




|?(z, t1)|2 |?(z, t2)|2


(c)




z z

Fig. 3.6. (a) z dependence of ?(r, t1) = u100(r)+ u200(r) for the time t1 = 2n?/?.
(b) z dependence of ?(r, t2) = u100(r)?u210(r) for t2 = (2n+1)?/? = t1 +?/?. (c) Corresponding dependencies of the probability densities |?(z, t1)|2 and |?(z, t2)|2 (with a slightly di?erent scale)


For the sake of simplicity, we ignore the spatial dependence altogether in the remainder of this section, and use
E(t) = E0 cos ?t . (3.113) This gives the interaction energy matrix element
Vab = ??E0 cos ?t . (3.114)

Substituting this into (3.83) with a as the ?nal state n and b as the initial state i, we have


|C(1) 2

2 sin2[(? ? ?)t/2]

? | = |?E0/2k|

(? ?

?)2/4

. (3.115)


This is the probability that the two-level atom absorbs energy under the in?uence of a driving ?eld, a phenomenon called (stimulated) absorption. Al-
ternatively, by identifying the initial state as a and the ?nal state as b, we
describe a process called stimulated emission. It is easy to show that |C(1)|2
in this case is the same as |C(1)|2 in the case of absorption: the probabili-
ties for stimulated emission and absorption are equal. This is an example of
microscopic reversibility.


Blackbody Radiation

Now consider the probability of a transition due to a ?eld that is not monochromatic, but rather has a continuous spectrum such as that for black-
body radiation. For this we replace the E2D(?) that occurs in using (3.115) by 2U ( )/?0, where U (?) is the energy density per radian/sec, and sum over all ?eld frequencies ?. The 2 here comes from the fact that two polarizations
are possible for each frequency. We then ?nd the total transition probability
¸ t2 sin2[(? ? ?)t/2]

PT =

d? U (?) k2

[(?

?)t/2]2 . (3.116)


This is the same kind of integral as that encountered for the Fermi Golden Rule in (3.84), except that for this two-level atom, we integrate over the ?eld
continuum frequency ? instead of the level continuum frequency ?. Here, we factor the slowly-varying energy density U (?) outside the integral, evaluating it at the peak, ? = ?, of the sin2 x/x2 curve. We ?nd the transition rate


? B(?)U (?) = 3k2?


?2U (?) , (3.117)


where the 3 comes from replacing ?2 by ?2/3 since the radiation can come from all directions: only 1/3 of the ?eld components e?ectively couple to the
dipole. Note that PT of (3.116) has the same value if the atom is initially in
the upper state, rather than in the lower state as taken for (3.1115). Hence
the stimulated emission rate equals the stimulated absorption rate of (3.117).
Another case in which a two-level atom interacts with a radiation con- tinuum is spontaneous emission, which can be described as a combination of radiation reaction (see end of Sect. 1.3) and stimulated emission by vacuum ?uctuations. This interpretation is clari?ed in Chap. 14, which derives the upper-level decay formula given by (3.89). For now just think of the radiation ?eld as consisting of a continuum of modes, each of which acts like a quan- tized simple harmonic oscillator. Section 3.4 shows that such an oscillator has a zero-point energy, which is associated with ?uctuations in the displacement variable. For the case of the electric ?eld, this displacement becomes the ?eld


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).