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,