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½¿½
Interaction of light with atoms

Even with the dipole approximation in force, the direct use of the atomic wave
function is completely impractical for a many-electron atom”this means any atom
with atomic number Z > 1. Fortunately, the complete description provided by the
many-electron wave function ψ (r1 , . . . , rZ ) is not needed. For the most part, only
selected properties”such as the discrete electronic energies and the matrix elements of
the dipole operator”are required. Furthermore these properties need not be calculated
ab initio; instead, they can be inferred from the measured wavelength and strength
of spectral lines. In this semi-empirical approach, the problem of atomic structure is
separated from the problem of the response of the atom to the electromagnetic ¬eld.
For a single atom interacting with the electromagnetic ¬eld, the discussion in Sec-
tion 4.2.1 shows that the state space is the tensor product H = HA — HF of the Hilbert
space HA for the atom and the Fock space HF for the ¬eld. A typical basis state for
H is |ψ, ¦ = |ψ |¦ , where |ψ and |¦ are respectively state vectors for the atom
and the ¬eld. Let us consider a typical matrix element ψ, ¦ |E (r)| ψ , ¦ of the elec-
tric ¬eld operator, where at least one of the vectors |ψ and |ψ describes a bound
state with characteristic spatial extent a, and |¦ and |¦ both describe states of the
¬eld containing only photons with wavelengths » a. On the scale of the optical
wavelengths, the atomic electrons can then be regarded as occupying a small region
surrounding the center-of-mass position,
Z
Mnuc Me
rcm = rnuc + rn , (4.125)
M M
n=1

where rcm is the operator for the center of mass, Me is the electron mass, rn is the
coordinate operator of the nth electron, Mnuc is the nuclear mass, rnuc is the coordinate
operator of the nucleus, Z is the atomic number, and M = Mnuc + ZMe is the total
mass.
For all practical purposes, the center of mass can be identi¬ed with the location
of the nucleus, since Mnuc ZMe . The plane-wave expansion (3.69) for the electric
¬eld then implies that the matrix element is slowly varying across the atom, so that
it can be expanded in a Taylor series around rcm ,

ψ, ¦ |E (r)| ψ , ¦
= ψ, ¦ |E (rcm )| ψ , ¦ + ψ, ¦ |[(r ’ rcm ) · ∇] E (rcm )| ψ , ¦ + · · · .
(4.126)
With the understanding that only matrix elements of this kind will occur, the
expansion can be applied to the ¬eld operator itself:

E (r) = E (rcm ) + [(r ’ rcm ) · ∇] E (rcm ) + · · · . (4.127)

The electric dipole approximation retains only the leading term in this expan-
sion, with errors of O (a/»). Keeping higher-order terms in the Taylor series incorpo-
rates successive terms in the general multipole expansion, e.g. magnetic dipole, electric
quadrupole, etc. In classical electrodynamics (Jackson, 1999, Sec. 4.2), the leading term
in the interaction energy of a neutral collection of charges with an external electric ¬eld
E is ’d · E, where d is the electric dipole moment. For an atom the dipole operator is
½¿¾ Interaction of light with matter

Z
(’e) (rn ’ rnuc ) .
d= (4.128)
n=1

Once again we rely on the correspondence principle to suggest that the interaction
Hamiltonian in the quantum theory should be

Hint = ’d · E (rcm ) . (4.129)

The atomic Hamiltonian can be expressed as
Z 2
P2 (pn )
Hatom = + + VC , (4.130)
2M n=1 2Me

Z Z
e2 Ze2
1 1

VC = , (4.131)
|rn ’ rl | 4π 0 |rn ’ rnuc |
4π 0 n=1
n=l=1

where VC is the Coulomb potential, P is the total momentum, and the pn s are a set of
relative momentum operators. Thus the Schr¨dinger-picture Hamiltonian in the dipole
o
approximation is H = Hem + Hatom + Hint .
The argument given in Section 4.2.2 shows that E (rcm ) is a hybrid operator acting
on both the atomic and ¬eld degrees of freedom. For most applications of quantum
optics, we can ignore this complication, since the De Broglie wavelength of the atom is
small compared to the interatomic spacing. In this limit, the center-of-mass position,
rcm , and the total kinetic energy P2 /2M can be treated classically, so that

P2
Hatom = + Hat , (4.132)
2M
where
Z 2
(pn )
Hat = + VC (4.133)
2Me
n=1

is the Hamiltonian for the internal degrees of freedom of the atom. In the same ap-
proximation, the interaction Hamiltonian reduces to

Hint = ’d · E (rcm ) , (4.134)

which acts jointly on the ¬eld states and the internal states of the atom.
In the rest frame of the atom, de¬ned by P = 0, the energy eigenstates

Hat |µq = µq |µq (4.135)

provide a basis for the Hilbert space, HA , describing the internal degrees of freedom
of the atom. The label q stands for a set of quantum numbers su¬cient to specify the
internal atomic state uniquely. The qs are discrete; therefore, they can be ordered so
that µq µq for q < q .
½¿¿
Interaction of light with atoms

In practice, the many-electron wave function ψq (r1 , . . . , rZ ) = r1 , . . . , rZ |µq can-
not be determined exactly, so the eigenstates are approximated, e.g. by using the
atomic shell model (Cohen-Tannoudji et al., 1977b, Chap. XIV, Complement A). In
this case the label q = (n, l, m) consists of the principal quantum number, the angular
momentum, and the azimuthal quantum number for the valence electrons in a shell
model description. The dipole selection rules are

= 0 unless l ’ l = ±1 and m ’ m = ±1, 0 .
µq d µq (4.136)

The z-axis is conventionally chosen as the quantization axis, and this implies

= 0 unless m ’ m = 0 ,
µq dz µq
(4.137)
= 0 unless m ’ m = ±1 .
µq dx µq = µq dy µq

A basis for the Hilbert space H = HA — HF describing the composite system of the
atom and the radiation ¬eld is given by the product vectors

|µq , n = |µq |n , (4.138)

where |n runs over the photon number states.
For a single atom the (c-number) kinetic energy P2 /2M can always be set to zero
by transforming to the rest frame of the atom, but when many atoms are present there
is no single frame of reference in which all atoms are at rest. Nevertheless, it is possible
to achieve a similar e¬ect by accounting for the recoil of the atom. Let us consider an
elementary process, e.g. absorption of a photon with energy ωk and momentum k by
an atom with energy µ1 + P2 /2M and momentum P. The ¬nal energy, µ2 + P 2 /2M ,
and momentum, P , are constrained by the conservation of energy,

ωk + µ1 + P2 /2M = µ2 + P 2 /2M , (4.139)

and conservation of momentum,

k+P=P . (4.140)

The initial and ¬nal velocities of the atom are respectively v = P/M and v = P /M ,
so eqn (4.140) tells us that the atomic recoil velocity is vrec = v ’ v = k/M .
Substituting P from eqn (4.140) into eqn (4.139) and expressing the result in terms
of vrec yields
1
ωk = ω21 + M vrec · v + vrec , (4.141)
2
where
µ2 ’ µ1
ω21 = (4.142)

is the Bohr frequency for this transition. For typical experimental conditions”e.g.
optical frequency radiation interacting with a tenuous atomic vapor”the thermal
½¿ Interaction of light with matter

velocities of the atoms are large compared to their recoil velocities, so that eqn (4.141)
can be approximated by
ω21
k·v,
ωk = ω21 + (4.143)
c
where k = k/k. Since v/c is small, this result can also be expressed as
ω21 = ωk ’ k · v . (4.144)
In other words, conservation of energy is equivalent to resonance between the atomic
transition and the Doppler shifted frequency of the radiation. With this thought
in mind, we can ignore the kinetic energy term in the atomic Hamiltonian and simply
tag each atom with its velocity and the associated resonance condition.
The next step is to generalize the single-atom results to a many-atom system. The
state space is now H = HA — HF , where the many-atom state space consists of product
(n) (n)
wave functions, i.e. HA = —n HA where HA is the (internal) state space for the nth
atom. Since Hint is linear in the atomic dipole moment, the part of the Hamiltonian
describing the interaction of the many-atom system with the radiation ¬eld is obtained
by summing eqn (4.129) over the atoms.
The Coulomb part is more complicated, since the general expression (4.131) con-
tains Coulomb interactions between charges belonging to di¬erent atoms. These inter-
atomic Coulomb potentials can also be described in terms of multipole expansions for
the atomic charge distributions. The interatomic potential will then be dominated by
dipole“dipole interactions. For tenuous vapors these e¬ects can be neglected, and the
many-atom Hamiltonian is approximated by H = Hem + Hat + Hint , where
(n)
Hat = Hat , (4.145)
n

Hint = ’ d(n) · E rcm ,
(n)
(4.146)
n
(n) (n)
and Hat , d(n) , and rcm are respectively the internal Hamiltonian, the electric dipole
operator, and the (classical) center-of-mass position for the nth atom.
4.9.2 The weak-¬eld limit
A second simpli¬cation comes into play for electromagnetic ¬elds that are weak, in the
sense that the dipole interaction energy is small compared to atomic energy di¬erences.
In other words |d · E| ωT , where d is a typical electric dipole matrix element, E is
a representative matrix element of the electric ¬eld operator, and ωT is a typical Bohr
frequency associated with an atomic transition. In terms of the characteristic Rabi
frequency
|d · E|
„¦= , (4.147)
which represents the typical oscillation rate of the atom induced by the electric ¬eld,
the weak-¬eld condition is
„¦ ωT . (4.148)

The Rabi frequency is given by „¦ = 1.39 — 107 d I, where „¦ is expressed in Hz, the
¬eld intensity I in W/cm2 , and the dipole moment d in debyes (1 D = 10’18 esu cm =
½¿
Interaction of light with atoms

0.33 — 10’29 C m). Typical values for the dipole matrix elements are d ∼ 1 D, and
the interesting Bohr frequencies are in the range 3 — 1010 Hz < ωT < 3 — 1015 Hz,
corresponding to wavelengths in the range 1 cm to 100 nm. For each value of ωT , eqn
(4.148) imposes an upper bound on the strength of the electric ¬elds associated with
the matrix elements of Hint . For a typical optical frequency, e.g. ωT ≈ 3 — 1014 Hz, the
upper bound is I ∼ 5—1014 W/cm2 , which could not be violated without vaporizing the
sample. At the long wavelength limit, » ∼ 1 cm (ωT ∼ 3 — 1010 Hz), the upper bound is
only I ∼ 5—106 W/cm2 , which could be readily violated without catastrophe. However
this combination of wavelength and intensity is not of interest for quantum optics,
since the corresponding photon ¬‚ux, 1029 photons/cm2 s, is so large that quantum
¬‚uctuations would be completely negligible. Thus in all relevant situations, we may
assume that the ¬elds are weak.
The weak-¬eld condition justi¬es the use of time-dependent perturbation theory
for the calculation of transition rates for spontaneous emission or absorption from an
incoherent radiation ¬eld. As we will see below, perturbation theory is not able to
describe other interesting phenomena, such as natural line widths and the resonant
coupling of an atom to a coherent ¬eld, e.g. a laser. Despite the failure of perturbation
theory for such cases, the weak-¬eld condition can still be used to derive a nonper-
turbative scheme which we will call the resonant wave approximation. Just as with
perturbation theory, the interaction picture is the key to understanding the resonant
wave approximation.

The Einstein A and B coe¬cients
4.9.3
As the ¬rst application of perturbation theory we calculate the Einstein A coe¬cient,
i.e. the total spontaneous emission rate for an atom in free space. For this and subse-
quent calculations, it will be convenient to write the interaction Hamiltonian as

Hint = ’ „¦(+) (r) + „¦(’) (r) , (4.149)

where the positive-frequency Rabi operator „¦(+) (r) is

E(+) (r) · d
(+)
„¦ (r) = , (4.150)

and r is the location of the atom. In the absence of boundaries, we can choose the
location of the atom as the origin of coordinates. Setting r = 0 in eqn (3.69) for
E(+) (r) and substituting into eqn (4.150) yields

ωk eks · d
„¦(+) = i aks . (4.151)
2 0V
ks

The initial state for the transition is |˜i = |µ2 , 0 = |µ2 |0 , where |µ2 is an
excited state of the atom and |0 is the vacuum state, so the initial energy is Ei = µ2 .
The ¬nal state is |˜f = |µ1 , 1ks = |µ1 |1ks , where |1ks = a† |0 is the state
ks
describing exactly one photon with wavevector k and polarization eks and |µ1 is an
atomic state with µ1 < µ2 . The ¬nal state energy is therefore Ef = µ1 + ωk . The
½¿ Interaction of light with matter

Feynman diagrams for emission and absorption are shown in Fig. 4.1. It is clear from
eqn (4.151) that only „¦(’) can contribute to emission, so the relevant matrix element
is
µ1 , 1ks „¦(’) µ2 , 0 = ’i„¦— (k) , (4.152)
21,s

where
ωk d21 · eks
„¦21,s (k) = (4.153)
2 0V
is the single-photon Rabi frequency for the 1 ” 2 transition, and d21 = µ2 d µ1
is the dipole matrix element. In the physical limit V ’ ∞, the photon energies ωk
become continuous, and the golden rule (4.113) can be applied to get the transition
rate
W1ks,2 = 2π |„¦21,s (k)|2 δ (ωk ’ ω21 ) . (4.154)
The irreversibility of the transition described by this rate is a mathematical con-
sequence of the continuous variation of the ¬nal photon energy that allows the use
of Fermi™s golden rule. A more intuitive explanation of the irreversible decay of an
excited atom is that radiation emitted into the cold and darkness of in¬nite space will
never return.
Since the spacing between discrete wavevectors goes to zero in the in¬nite volume
limit, the physically meaningful quantity is the emission rate into an in¬nitesimal k-
space volume d3 k centered on k. For each polarization, the number of k-modes in d3 k
3
is V d3 k/ (2π) ; consequently, the di¬erential emission rate is

V d3 k
dW1ks,2 = W1ks,2 3
(2π)
d3 k
2
= 2π |M21,s (k)| δ (ωk ’ ω21 ) , (4.155)
(2π)3

where
√ ωk d21 · eks
M21,s (k) = V „¦21,s (k) = . (4.156)
20
The Einstein A coe¬cient is the total transition rate into all ks-modes:



µ
kI
µ


µ
kI
Fig. 4.1 First-order Feynman diagrams for
µ
emission (1) and absorption (2). Straight lines
correspond to atomic states and wiggly lines
 
to photon states.
½¿
Interaction of light with atoms

d3 k 2
2π |M21,s (k)| δ (ωk ’ ω21 ) .
A2’1 = (4.157)
3
(2π) s

The integral over the magnitude of k can be carried out by the change of variables
k ’ ω/c. It is customary to write this result in terms of the density of states, D (ω21 ),
which is the number of resonant modes per unit volume per unit frequency. The number
of modes in d3 k is 2V d3 k/ (2π)3 , where the factor 2 counts the polarizations for each
k, so the density of states is

d3 k 2
ω21
D (ω21 ) = 2 3 δ (ωk ’ ω21 ) = π 2 c3 . (4.158)
(2π)

This result includes the two polarizations and the total 4π sr of solid angle, so calcu-
lating the contribution from a single plane wave requires division by 8π. In this way
A2’1 is expressed as an average over emission directions and polarizations,

d„¦k 1 2
2π |M21,s (k)| D (ω21 ) ,
A2’1 = (4.159)
4π 2 s

where d„¦k = sin (θk ) dθk dφk . The average over polarizations is done by using eqn
(4.153) and the completeness relation (B.49) to get

1 1
(di )21 (dj )— eksi e—
|d21 · eks |2 = ksj
21
2 2
s s

1—
d · d21 ’ k · d21 k · d21
= . (4.160)
2 21
In some cases the vector d21 is real, but this cannot be guaranteed in general (Mandel
and Wolf, 1995, Sec. 15.1.1). When d21 is complex it can be expressed as d21 =
d21 + id21 , where d21 and d21 are both real vectors. Inserting this into the previous
equation gives
2 2
2 2
2
|d21 · eks | = (d21 ) ’ k · d21 + (d21 ) ’ k · d21 , (4.161)
s

and the remaining integral over the angles of k can be carried out for each term by
choosing the z-axis along d21 or d21 . The result is

4 |d21 |2 k0
3
1
A2’1 = , (4.162)
4π 0 3

where k0 = ω21 /c = 2π/»0 and |d21 | = d— · d21 . This agrees with the value ob-
2
21
tained earlier by Einstein™s thermodynamic argument. Dropping the coe¬cient in
square brackets gives the result in Gaussian units.
Einstein™s quantum model for radiation involves two other coe¬cients, B1’2 for
absorption and B2’1 for stimulated emission. The stimulated emission rate is the rate
½¿ Interaction of light with matter

for the transition |µ2 , nks ’ |µ1 , nks + 1 , i.e. the initial state has nks photons in the
mode ks. In this case eqn (4.152) is replaced by

µ1 , nks + 1 „¦(’) µ2 , nks = ’i„¦— (k) nks + 1 , (4.163)
21,s



where the factor nks + 1 comes from the rule a† |n = n + 1 |n + 1 . For nks = 0,

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