= 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,

this reduces to the spontaneous emission result, so the only di¬erence between the two

√

processes is the enhancement factor nks + 1. In order to simplify the argument we

will assume that nks = n (ω), i.e. the photon population is independent of polarization

and propagation direction. Then the average over polarizations and emission directions

produces

“ = [n (ω21 ) + 1] A2’1 = A2’1 + n (ω21 ) A2’1 , (4.164)

where the two terms are the spontaneous and stimulated rates respectively. By com-

paring this to eqn (1.13), we see that B2’1 ρ (ω21 ) = n (ω21 ) A2’1 , where ρ (ω21 ) is

the energy density per unit frequency. In the present case this is

3

ω21

ρ (ω21 ) = ( ω21 ) n (ω21 ) D (ω21 ) = 2 3 n (ω21 ) , (4.165)

πc

so the relation between the A and B coe¬cients is

3

A2’1 ω21

= 2 3, (4.166)

B2’1 πc

in agreement with eqn (1.21). The absorption coe¬cient B1’2 is deduced by calculat-

ing the transition rate for |µ1 , nks + 1 ’ |µ2 , nks . The relevant matrix element,

√

µ2 , nks „¦(+) µ1 , nks + 1 = i„¦21,s (k) nks + 1 , (4.167)

corresponds to part (2) of Fig. 4.1. Since |„¦21,s (k)| = „¦— (k) , using this matrix

21,s

element in eqn (4.113) will give the same result as the calculation of the stimulated

emission coe¬cient, therefore the absorption rate is identical to the stimulated emission

rate, i.e. B1’2 = B2’1 , in agreement with the detailed-balance argument eqn (1.18).

Thus the quantum theory correctly predicts the relations between the Einstein A and

B coe¬cients, and it provides an a priori derivation for the spontaneous emission rate.

Spontaneous emission in a planar cavity—

4.9.4

One of the assumptions in Einstein™s quantum model for radiation is that the A and

B coe¬cients are solely properties of the atom, but further thought shows that this

cannot be true in general. Consider, for example, an atom in the interior of an ideal

cubical cavity with sides L. According to eqn (2.15) the eigenfrequencies satisfy ωn

√

2πc/L; therefore, resonance is impossible if the atomic transition frequency is too

√ √

small, i.e. ω21 < 2πc/L, or equivalently L < »0 / 2, where »0 = 2πc/ω21 is the

wavelength of the emitted light. In addition to this failure of the resonance condition,

the golden rule (4.113) is not applicable, since the mode spacing is not small compared

to the transition frequency.

½¿

Interaction of light with atoms

What this means physically is that photons emitted by the atom are re¬‚ected from

the cavity walls and quickly reabsorbed by the atom. This behavior will occur for

any ¬nite value of L, but clearly the minimum time required for the radiation to be

reabsorbed will grow with L. In the limit L ’ ∞ the time becomes in¬nite and the

result for an atom in free space is recovered. Therefore the standard result (4.162) for

A2’1 is only valid for an atom in unbounded space.

The fact that the spontaneous emission rate for atoms is sensitive to the bound-

ary conditions satis¬ed by the electromagnetic ¬eld was recognized long ago (Purcell,

1946). More recently this problem has been studied in conjunction with laser etalons

(Stehle, 1970) and materials exhibiting an optical bandgap (Yablonovitch, 1987). We

will illustrate the modi¬cation of spontaneous emission in a simple case by describ-

ing the theory and experimental results for an atom in a planar cavity of the kind

considered in connection with the Casimir e¬ect.

A Theory

For this application, we will assume that the transverse dimensions are large, L »0 ,

while the longitudinal dimension ∆z (along the z-axis) is comparable to the transition

wavelength, ∆z ∼ »0 . The mode wavenumbers are then k = q + (nπ/∆z) uz , where

q = kx ux + ky uy , and the cavity frequencies are

2 1/2

nπ

2

ωqn =c q + . (4.168)

∆z

Both n and q are discrete, but the transverse mode numbers q will become densely

spaced in the limit L ’ ∞. The Schr¨dinger-picture ¬eld operator is given by the

o

analogue of eqn (3.69),

Cn

ωqn

aqns E qns (r) ,

(+)

E (r) = i (4.169)

20

q n s=1

where the mode functions are described in Appendix B.4 and Cn is the number of

independent polarization states for the mode (n, q): C0 = 1 and Cn = 2 for n 1.

Since the separation, ∆z, between the plates is comparable to the wavelength, the

transition rate will depend on the distance from the atom to each plate. Consequently,

we are not at liberty to assume that the atom is located at any particular z-value. On

the other hand, the dimensions along the x- and y-axes are e¬ectively in¬nite, so we

can choose the origin in the (x, y)-plane at the location of the atom, i.e. r = (0, z). The

interaction Hamiltonian is given by eqns (4.149) and (4.150), but the Rabi operator

in this case is a function of z, with the positive-frequency part

Cn

ωqn

aqns d · E qns (0, z) .

(+)

„¦ (z) = i (4.170)

20

q n s=1

The transition of interest is |µ2 , 0 ’ |µ1 , 1qns , so only „¦(’) (z) can contribute.

For each value of n and z the remaining calculation is a two-dimensional version of

½¼ Interaction of light with matter

the free-space case. Substituting the relevant matrix elements into eqn (4.113) and

multiplying by L2 d2 q/ (2π)2 ”the number of modes in the wavevector element d2 q”

yields the di¬erential transition rate

d2 q

2

dW2’1,qns (z) = 2π |M21,ns (q, z)| δ (ω21 ’ ωqn ) , (4.171)

(2π)2

where

ωqn

Ld21 · E qns (0, z) .

M21,ns (q, z) = (4.172)

20

For a given n, the transition rate into all transverse wavevectors q and polarizations

s is

Cn

d2 q 2

2 2π |M21,ns (q, z)| δ (ω21 ’ ωqn ) ,

A2’1,n (z) = (4.173)

(2π)

s=1

and the total transition rate is the sum of the partial rates for each n,

∞

A2’1 (z) = A2’1,n (z) . (4.174)

n=0

The delta function in eqn (4.173) is eliminated by using polar coordinates, d2 q =

qdqdφ, and then making the change of variables q ’ ω/c = ωqn /c. The result is

customarily expressed in terms of a density of states factor Dn (ω21 ), de¬ned as the

number of resonant modes per unit frequency per unit of transverse area. For a given

n there are Cn polarizations, so

d2 q

Dn (ω21 ) = Cn δ (ω21 ’ ωqn )

(2π)2

dω ω

Cn

δ (ω21 ’ ω)

=

2π ω0n c2

Cn ω21 n»0

θ ∆z ’

= , (4.175)

2πc2 2

where θ (ν) is the standard step function, »0 = 2πc/ω21 is the wavelength for the

transition, and ω0n = nπc/∆z. This density of states counts all polarizations and the

full azimuthal angle, so in evaluating eqn (4.173) the extra 2πCn must be divided out.

The transition rate then appears as an average over azimuthal angles and polarizations:

Cn

1 dφ 2

A2’1,n (z) = Dn (ω21 ) 2π |M21,qns (z)| . (4.176)

Cn 2π

s=1

According to eqn (4.175) the density of states vanishes for ∆z/»0 < n/2; therefore,

emission into modes with n > 2∆z/»0 is forbidden. This re¬‚ects the fact that the

high-n modes are not in resonance with the atomic transition. On the other hand,

the density of states for the (n = 0)-mode is nonzero for any value of ∆z/»0 , so this

½½

Interaction of light with atoms

transition is only forbidden if it violates atomic selection rules. In fact, this is the only

possible decay channel for ∆z < »0 /2. In this case the total decay rate is

2πk0 |(dz )21 |2 3 |(dz )21 |2

2

1 »0

= = Avac ,

A2’1,0 (4.177)

2

|d21 |

4π 0 ∆z 4∆z

where Avac is the vacuum value given by eqn (4.162). The factor in square brackets

is typically of order unity, so the decay rate is enhanced over the vacuum value when

∆z < »0 /4, and suppressed below the vacuum value for »0 /4 < ∆z < »0 /2.

If the dipole selection rules (4.137) impose (dz )21 = 0, then decay into the (n = 0)-

mode is forbidden, and it is necessary to consider somewhat larger separations, e.g.

»0 /2 < ∆z < »0 . In this case, the decay to the (n = 1)-mode is the only one allowed.

There are now two polarizations to consider, the P -polarization in the (q, uz )-plane

and the orthogonal S-polarization along uz — q. We will simplify the calculation by

assuming that the matrix element d21 is real. In the general case of complex d21 a

separate calculation for the real and imaginary parts must be done, as in eqn (4.161).

For real d21 the polar angle φ can be taken as the angle between d21 and q. The

assumption that (dz )21 = 0 combines with the expressions (B.82) and (B.83), for the

P - and S-polarizations, respectively, to yield

2

3 »0 »0 πz »0

θ ∆z ’

sin2

A2’1,1 = 1+ Avac , (4.178)

2 2∆z 2∆z ∆z 2

2

where we have used the selection rule to impose d⊥ = d2 . The decay rate depends

on the location of the atom between the plates, and achieves its maximum value at

the midplane z = ∆z/2. In a real experiment, there are many atoms with unknown

locations, so the observable result is the average over z:

2

3 »0 »0 »0

θ ∆z ’

A2’1,1 = 1+ Avac . (4.179)

4 2∆z 2∆z 2

This rate vanishes for »0 > 2∆z, and for »0 /2∆z slightly less than unity it is enhanced

over the vacuum value:

3

A2’1,1 Avac for »0 /2∆z 1. (4.180)

2

The decay rate is suppressed below the vacuum value for »0 /2∆z 0.8.

B Experiment

The clear-cut and striking results predicted by the theoretical model are only possible

if the separation between the plates is comparable to the wavelength of the emitted

radiation. This means that experiments in the optical domain would be extremely

di¬cult. The way around this di¬culty is to use a Rydberg atom, i.e. an atom which

has been excited to a state”called a Rydberg level”with a large principal quantum

number n. The Bohr frequencies for dipole allowed transitions between neighboring

½¾ Interaction of light with matter

high-n states are of O 1/n3 , so the wavelengths are very large compared to optical

wavelengths.

In the experiment we will discuss here (Hulet et al., 1985), cesium atoms were

excited by two dye laser pulses to the |n = 22, m = 2 state. The small value of

the magnetic quantum number is explained by the dipole selection rules, ∆l = ±1,

∆m = 0, ±1. These restrictions limit the m-values achievable in the two-step exci-

tation process to a maximum of m = 2. This is a serious problem, since the state

|n = 22, m = 2 can undergo dipole allowed transitions to any of the states |n , m for

2n 21 and m = 1, 3. A large number of decay channels would greatly compli-

cate both the experiment and the theoretical analysis. This complication is avoided by

exposing the atom to a combination of rapidly varying electric ¬elds and microwave

radiation which leave the value of n unchanged, but increase m to the maximum pos-

sible value, m = n ’ 1, a so-called circular state that corresponds to a circular Bohr

orbit. The overall process leaves the atom in the state |n = 22, m = 21 which can

only decay to |n = 21, m = 20 . This simpli¬es both the experimental situation and

the theoretical model. The wavelength for this transition is »0 = 0.45 mm, so the me-

chanical problem of aligning the parallel plates is much simpler than for the Casimir

force experiment. The gold-plated aluminum plates are held apart by quartz spacers

at a separation of ∆z = 230.1 µm so that »0 /2∆z = 0.98.

The atom has now been prepared so that there is only one allowed atomic transi-

tion, but there are still two modes of the radiation ¬eld, E q0 and E q1s , into which the

atom can decay. There is also the di¬cult question of how to produce controlled small

changes in the plate spacing in order to see the e¬ects on the spontaneous emission

rate. Both of these problems are solved by the expedient of establishing a voltage drop

between the plates. The resulting static electric ¬eld polarizes the atom so that the

natural quantization axis lies in the direction of the ¬eld. The matrix elements of the

z-component of the dipole operator, m dz m , vanish unless m = m, but transitions

of this kind are not allowed by the dipole selection rules, m = m ± 1, for the circu-

lar Rydberg atom. This amounts to setting (dz )21 = 0. Emission of E q0 -photons is

therefore forbidden, and the atom can only emit E q1s -photons. The ¬eld also causes

second-order Stark shifts (Cohen-Tannoudji et al., 1977b, Complement E-XII) which

decrease the di¬erence in the atomic energy levels and thus increase the wavelength

»0 . This means that the wavelength can be modi¬ed by changing the voltage, while the

plate spacing is left ¬xed. The onset of ¬eld ionization limits the ¬eld strength that can

be employed, so the wavelength can only be tuned by ∆» = 0.04 »0 . Fortunately, this

is su¬cient to increase the ratio »0 /2∆z through the critical value of unity, at which

the spontaneous emission should be quenched. At room temperature the blackbody

spectrum contains enough photons at the transition frequency to produce stimulated

emission. The observed emission rate would then be the sum of the stimulated and

spontaneous decay rates. In the model this would mean that we could not assume that

the initial state is |µ1 , 0 . This additional complication is avoided by maintaining the

apparatus at 6.5 K. At this low temperature, stimulated emission due to blackbody

radiation at »0 is strongly suppressed.

A thermal atomic beam of cesium ¬rst passes through a production region, where

the atoms are transferred to the circular state, then through a drift region”of length

½¿

Interaction of light with atoms

L = 12.7 cm”between the parallel plates. The length L is chosen so that the mean

transit time is approximately the same as the vacuum lifetime. After passing through

the drift region the atoms are detected by ¬eld ionization in a region where the ¬eld

increases with length of travel. The ionization rates for n = 22 and n = 21 atoms di¬er

substantially, so the location of the ionization event allows the two sets of atoms to be

resolved.

In this way, the time-of-¬‚ight distribution of the n = 22 atoms was measured. In

the absence of decay, the distribution would be determined by the original Boltzmann

distribution of velocities, but when decay due to spontaneous emission is present, only

the faster atoms will make it through the drift region. Thus the distribution will shift

toward shorter transit times. In the forbidden region, »0 /2∆z > 1, the data were

consistent with A2’1,1 = 0, with estimated errors ±0.05Avac . In other words, the

lifetime of an atom between the plates is at least twenty times longer than the lifetime

of the same atom in free space.

Raman scattering—

4.9.5

In Raman scattering, a photon at one frequency is absorbed by an atom or molecule,

and a photon at a di¬erent frequency is emitted. The simplest energy-level diagram

permitting this process is shown in Fig. 4.2. This is a second-order process, so it

(2)

requires the calculation of the second-order amplitude Vf i , where the initial and

¬nal states are respectively |˜i = |µ1 , 1ks and |˜f = |µ2 , 1k s . The representation

(4.149) allows the operator product on the right side of eqn (4.120) to be written as

2

„¦(’) (t1 ) „¦(’) (t2 ) + „¦(+) (t1 ) „¦(+) (t2 )

Hint (t1 ) Hint (t2 ) =

2

„¦(’) (t1 ) „¦(+) (t2 ) + „¦(+) (t1 ) „¦(’) (t2 ) ,

+

(4.181)

where the ¬rst two terms change photon number by two and the remaining terms

leave photon number unchanged. Since the initial and ¬nal states have equal photon

number, only the last two terms can contribute in eqn (4.120); consequently, the matrix

element of interest is

2

˜f „¦(’) (t1 ) „¦(+) (t2 ) + „¦(+) (t1 ) „¦(’) (t2 ) ˜i . (4.182)

!

k I

kI

Fig. 4.2 Raman scattering from a three-level

atom. The transitions 1 ” 3 and 2 ” 3 are

dipole allowed. A photon in mode ks scatters

into the mode k s .

½ Interaction of light with matter

Since t2 < t1 the ¬rst term describes absorption of the initial photon followed by

emission of the ¬nal photon, as one would intuitively expect. The second term is rather

counterintuitive, since the emission of the ¬nal photon precedes the absorption of the

initial photon. These alternatives are shown respectively by the Feynman diagrams

(1) and (2) in Fig. 4.3, which we will call the intuitive and counterintuitive diagrams

respectively.

The calculation of the transition amplitude by eqn (4.123) yields

µ2 , 1k s „¦(’) Λu Λu „¦(+) µ1 , 1ks

(2)

= ’i 2πδ (ωk ’ ωk ’ ω21 )

Vf i µ2 ’Eu

ωk + +i

u

µ2 , 1k s „¦(+) Λu Λu „¦(’) µ1 , 1ks

’i 2πδ (ωk ’ ωk ’ ω21 ) ,

µ2 ’Eu

ωk + +i

u

(4.183)

where the two sums over intermediate states correspond respectively to the intuitive

and counterintuitive diagrams. Since „¦(+) decreases the photon number by one, the

intermediate states in the ¬rst sum have the form |Λu = |µq , 0 . In this simple model

the only available state is |Λu = |µ3 , 0 . Thus the energy is Eu = µ3 and the denomi-

nator is ωk ’ω32 +i . In fact, the intermediate state can be inferred from the Feynman

diagram by passing a horizontal line between the two vertices. For the intuitive dia-

gram, the only intersection is with the internal atom line, but in the counterintuitive

diagram the line passes through both photon lines as well as the atom line. In this

case, the intermediate state must have the form |Λu = |µ1 , 1ks , 1k s , with energy

Eu = µ3 + ωk + ωk and denominator ’ωk ’ ω32 + i . These claims can be veri¬ed

by a direct calculation of the matrix elements in the second sum.

This calculation yields the explicit expression

— —

M32,s (k ) M31,s (k) M23,s (k) M13,s (k ) 2π

(2)

= ’i δ (ωk ’ ωk ’ ω21 ) ,

Vf i +

ωk ’ ω32 + i ’ωk ’ ω32 + i V

(4.184)

k I

3

kI

3

k I

Fig. 4.3 Feynman diagrams for Raman scat-

kI

tering. Diagram (1) shows the intuitive order-

ing in which the initial photon is absorbed

prior to the emission of the ¬nal photon. Di-

agram (2) shows the counterintuitive case in

which the order is reversed.

½

Exercises

(2) 2

where the matrix elements are de¬ned in eqn (4.156). Multiplying Vf i by the

number of modes V d3 k/ (2π)3 V d3 k / (2π)3 and using the rule (4.119) gives the

di¬erential transition rate

— — 2

M32,s (k ) M31,s (k) M23,s (k) M13,s (k )

dW3ks’2k s = 2π +

ωk ’ ω32 + i ’ωk ’ ω32 + i

d3 k d3 k

— δ (ωk ’ ωk ’ ω21 ) 3. (4.185)

3

(2π) (2π)

4.10 Exercises

4.1 Semiclassical electrodynamics

(1) Derive eqn (4.7) and use the result to get eqn (4.27).

(2) For the classical ¬eld described in the radiation gauge, do the following.

(a) Derive the equation satis¬ed by the scalar potential • (r).

(b) Show that

1

∇2 = ’4πδ (r ’ r0 ) .

|r ’ r0 |

(c) Combine the last two results to derive the Coulomb potential term in eqn

(4.31).

4.2 Maxwell™s equations from the Heisenberg equations of motion

Derive Maxwell™s equations and Lorentz equations of motion as given by eqns (4.33)“

(4.37), and eqn (4.42), using Heisenberg™s equations of motions and the relevant equal-

time commutators.

Spatial inversion and time reversal—

4.3

(1) Use eqn (4.55) to evaluate UP |n for a general number state, and explain how to

extend this to all states of the ¬eld.

(2) Verify eqn (4.61) and ¬ll in the details needed to get eqn (4.64).

(3) Evaluate ΛT |n for a general number state, and explain how to extend this to all

states of the ¬eld. Watch out for antilinearity.

4.4 Stationary density operators

Use eqns (3.83), (4.67), and U (’t) = U † (t), together with cyclic invariance of the

trace, to derive eqns (4.69) and (4.71).

4.5 Spin-¬‚ip transitions

The neutron is a spin-1/2 particle with zero charge, but it has a nonvanishing magnetic

moment MN = ’ |gN | µN σ, where gN is the neutron gyromagnetic ratio, µN is the

nuclear magneton, and σ = (σx , σy , σz ) is the vector of Pauli matrices. Since the

neutron is a massive particle, it is a good approximation to treat its center-of-mass

½ Interaction of light with matter

motion classically. All of the following calculations can, therefore, be done assuming

that the neutron is at rest at the origin.

(1) In the presence of a static, uniform, classical magnetic ¬eld B0 the Schr¨dinger-

o

picture Hamiltonian”neglecting the radiation ¬eld”is H0 = ’MN · B0 . Take the

z-axis along B0 , and solve the time-independent Schr¨dinger equation, H0 |ψ =

o

µ |ψ , for the ground state |µ1 , the excited state |µ2 , and the corresponding en-

ergies µ1 and µ2 .

(2) Include the e¬ects of the radiation ¬eld by using the Hamiltonian H = H0 + Hint,

where Hint = ’MN · B and B is given by eqn (3.70), evaluated at r = 0.

(a) Evaluate the interaction-picture operators aks (t) and σ± (t) in terms of the

Schr¨dinger-picture operators aks and σ± = (σx ± iσy ) /2 (see Appendix

o

C.3.1). Use the results to ¬nd the time dependence of the Cartesian com-

ponents σx (t), σy (t), σz (t).

(b) Find the condition on the ¬eld strength |B0 | that guarantees that the zero-

order energy splitting is large compared to the strength of Hint , i.e.

µ2 ’ µ1 | µ1 , 1ks |Hint | µ2 , 0 | ,

where |µ1 , 1ks = |µ1 |1ks , |µ2 , 0 = |µ2 |0 , and |1ks = a† |0 . Explain the

ks

physical signi¬cance of this condition.

(c) Using Section 4.9.3 as a guide, calculate the spontaneous emission rate (Ein-

stein A coe¬cient) for a spin-¬‚ip transition. Look up the numerical values

of |gN | and µN and use them to estimate the transition rate for magnetic

¬eld strengths comparable to those at the surface of a neutron star, i.e.

|B0 | ∼ 1012 G.

4.6 The quantum top

Replace the unperturbed Hamiltonian in Exercise 4.5 by H0 = ’MN · B0 (t), where

B0 (t) changes direction as a function of time. Use this Hamiltonian to derive the

Heisenberg equations of motion for σ (t) and show that they can be written in the

same form as the equations for a precessing classical top.

4.7 Transition probabilities for a neutron in combined static and

radio-frequency ¬elds—

Solve the Schr¨dinger equation for a neutron in a combined static and radio-frequency

o

magnetic ¬eld. A static ¬eld of strength B0 is applied along the z-axis, and a circularly-

polarized, radio-frequency ¬eld of classical amplitude B1 and frequency ω is applied

in the (x, y)-plane, so that the total Hamiltonian is H = H0 + Hint , where

H0 = ’Mz B0 ,

Hint = ’MxB1 cos ωt + My B1 sin ωt ,

Mx = 1 µσx , My = 1 µσy , Mz = 1 µσz , µ is the magnetic moment of the neutron,

2 2 2

and the σs are Pauli matrices. Show that the probability for a spin ¬‚ip of the neutron

initially prepared (at t = 0) in the ms = + 1 state to the ms = ’ 1 state is given by

2 2

½

Exercises

1

P 1 ’’ 1 (t) = sin2 ˜ sin2 at ,

2

2 2

where

2

ω1

sin2 ˜ = ,

(ω0 ’ ω)2 + ω1

2

2

(ω0 ’ ω) + ω1 ,

2

a=

ω0 = µB0 / , and ω1 = µB1 / . Interpret this result geometrically (Rabi et al., 1954).

5

Coherent states

In the preceding chapters, we have frequently called upon the correspondence principle

to justify various conjectures, but we have not carefully investigated the behavior of

quantum states in the correspondence-principle limit. The di¬culties arising in this

investigation appear in the simplest case of the excitation of a single cavity mode

E κ (r). In classical electromagnetic theory”as described in Section 2.1”the state of

a single mode is completely described by the two real numbers (Qκ0 , Pκ0 ) specifying

the initial displacement and momentum of the corresponding radiation oscillator. The

subsequent motion of the oscillator is determined by Hamilton™s equations of motion.

The set of classical ¬elds representing excitation of the mode κ is therefore represented

by the two-dimensional phase space {(Qκ , Pκ )}.

In striking contrast, the quantum states for a single mode belong to the in¬nite-

dimensional Hilbert space spanned by the family of number states, {|n , n = 0, 1, . . .}.

In order for a state |Ψ to possess a meaningful correspondence-principle limit, each

member of the in¬nite set, {cn = n |Ψ , n = 0, 1, . . .}, of expansion coe¬cients must

be expressible as a function of the two classical degrees of freedom (Qκ0 , Pκ0 ). This

observation makes it clear that the number-state basis is not well suited to demonstrat-

ing the correspondence-principle limit. In addition to this fundamental issue, there are

many applications for which a description resembling the classical phase space would

be an advantage.

These considerations suggest that we should search for quantum states of light that

are quasiclassical; that is, they approach the classical description as closely as possi-

ble. To this end, we ¬rst review the solution of the corresponding problem in ordinary

quantum mechanics, and then apply the lessons learnt there to the electromagnetic

¬eld. After establishing the basic form of the quasiclassical states, we will investigate

possible physical sources for them and the experimental evidence for their existence.

The ¬nal sections contain a review of the mathematical properties of quasiclassical

states, and their use as a basis for representations of general quantum states.

5.1 Quasiclassical states for radiation oscillators

In order to simplify the following discussion, we will at ¬rst only consider situations

in which a single mode of the electromagnetic ¬eld is excited. For example, excitation

of the mode E κ (r) in an ideal cavity corresponds to the classical ¬elds

½

Quasiclassical states for radiation oscillators

1

A (r, t) = √ Qκ (t) E κ (r) ,

0

(5.1)

1

E (r, t) = ’ √ Pκ (t) E κ (r) .

0

5.1.1 The mechanical oscillator

In Section 2.1 we guessed the form of the quantum theory of radiation by using the

mathematical identity between a radiation oscillator and a mechanical oscillator of

unit mass. The real Q and P variables of the classical oscillator can be simultaneously

speci¬ed; therefore, the trajectory (Q (t) , P (t)) of the oscillator is completely described

by the time-dependent, complex amplitude

ωQ (t) + iP (t)

√

A (t) = , (5.2)

2ω

where the is introduced for dimensional convenience only. Hamilton™s equations of

motion for the real variables Q and P are equivalent to the complex equation of motion

™

A = ’iωA , (5.3)

with the general solution given by the phasor (a complex number of ¬xed modulus)

A (t) = ± exp (’iωt) . (5.4)

The initial complex amplitude of the oscillator is related to ± by

ωQ0 + iP0

√

A (t = 0) = = ±, (5.5)

2ω

and the conserved classical energy is

1 22

ω Q0 + P0 = ω±— ± .

Ecl = (5.6)

2

Taking the real and imaginary parts of A (t), as given in eqn (5.4), shows that the

solution traces out an ellipse in the (Q, P ) phase space. An equivalent representation

is the circle traced out by the tip of the phasor A (t) in the complex (Re A, Im A)

space.

For the quantum oscillator, the classical amplitude A (0) and the energy ω |±|2

are respectively replaced by the lowering operator

ωq + ip

a= √ (5.7)

2ω

and the Hamiltonian operator Hosc = ωa† a. The Heisenberg equation of motion for

a(t),

da i

= ’ [a, Hosc ] = ’iωa , (5.8)

dt

has the same form as the classical equation of motion (5.3).

½¼ Coherent states

We can now use an argument from quantum mechanics (Cohen-Tannoudji et al.,

1977a, Chap. V, Complement G) to construct the quasiclassical state. According to

the correspondence principle, the classical quantities ± and Ecl must be identi¬ed with

the expectation values of the corresponding operators, so the quasiclassical state |φ

corresponding to the classical value ± should satisfy φ |a| φ = ± and φ Hosc φ =

Ecl = ω |±|2 . Inserting Hosc = ωa† a into the latter condition and using the former

2

condition to evaluate |±| produces

φ a † a φ = φ a† φ φ |a| φ . (5.9)

The joint variance of two operators X and Y , de¬ned by

V (X, Y ) = (X ’ X ) (Y ’ Y ) = XY ’ X Y, (5.10)

reduces to the ordinary variance V (X) for X = Y . In this language, the meaning of

eqn (5.9) is that the joint variance of a and a† vanishes,

V a† , a = 0 , (5.11)

i.e. the operators a and a† are statistically independent for a quasiclassical state. In

its present form it is not obvious that V a† , a refers to measurable quantities, but

this concern can be addressed by using eqn (5.7) to get the equivalent form

ω 1 1

V a† , a =

2 2

(q ’ q ) (p ’ p ) ’ .

+ (5.12)

2 2ω 2

The condition (5.11) is the fundamental property de¬ning quasiclassical states, and

it determines |φ up to a phase factor. To see this, we de¬ne a new operator b = a ’ ±

and a new state |χ = b |φ , to get

χ| χ = φ b† b φ = V a† , a = 0 . (5.13)

The squared norm χ |χ only vanishes if |χ = 0; consequently, a |φ = ± |φ . Thus

the quasiclassical state |φ is an eigenstate of the lowering operator a with eigenvalue

±. For this reason it is customary to rename |φ as |± , so that

a |± = ± |± . (5.14)

For non-hermitian operators, there is no general theorem guaranteeing the existence

of eigenstates, so we need to ¬nd an explicit solution of eqn (5.14). In this section,

we will do this in the usual coordinate representation, in order to gain an intuitive

understanding of the physical signi¬cance of |± . In the following section, we will ¬nd

an equivalent form by using the number-state basis. This is useful for understanding

the statistical properties of |± .

The coordinate-space wave function for |± is φ± (Q) = Q| ± , where q |Q =

Q |Q . In this representation, the action of q is qφ± (Q) = Qφ± (Q), and the action of

½½

Quasiclassical states for radiation oscillators

the momentum operator is pφ± (Q) = ’i (d/dQ) φ± (Q). After inserting this into eqn

(5.7), the eigenvalue problem (5.14) is represented by the di¬erential equation

1 d

√ ωQ + φ± (Q) = ±φ± (Q) , (5.15)

dQ

2ω

which has the normalizable solution

2

(Q ’ Q0 )

1/4

ω P0 Q

exp ’

φ± (Q) = exp i (5.16)

2

π 4∆q0

for any value of the complex parameter ±. The parameters Q0 and P0 are given by

√

Q0 = 2 /ω Re ±, P0 = 2 ω Im ±, and the width of the Gaussian is ∆q0 = /2ω.

We have chosen the prefactor so that φ± (Q) is normalized to unity. For Q0 = P0 =

0, φ0 (Q) is the ground-state wave function of the oscillator; therefore, the general

quasiclassical state, φ± (Q), represents the ground state of an oscillator which has been

displaced from the origin of phase space to the point (Q0 , P0 ). For the Q dependence

this is shown explicitly by the probability density |φ± (Q)|2 , which is a Gaussian in

Q centered on Q0 . An alternative representation using the momentum-space wave

function, φ± (P ) = P | ± , can be derived in the same way”or obtained from φ± (Q)

by Fourier transform”with the result

2

(P ’ P0 ) Q0 P

’1/4

exp ’ exp ’i

φ± (P ) = (π ω) , (5.17)

4∆p20

ω/2. The product ∆p0 ∆q0 = /2, so |± is a minimum-uncertainty

where ∆p0 =

state; it is the closest we can come to the classical description. The special values

∆q0 = /2ω and ∆p0 = ω/2 de¬ne the standard quantum limit for the

harmonic oscillator.

5.1.2 The radiation oscillator

Applying these results to the radiation oscillator for a particular mode E κ involves a

change of terminology and, more importantly, a change in physical interpretation. For

the radiation oscillator corresponding to the mode E κ , the de¬ning equation (5.14) for

a quasiclassical state is replaced by

aκ |±κ = δκ κ ±κ |±κ ; (5.18)

in other words, the quasiclassical state for this mode is the vacuum state for all other

modes. This is possible because the annihilation operators for di¬erent modes commute

with each other. A simple argument using eqn (5.18) shows that the averages of all

normal-ordered products completely factorize:

m

a† ±κ = (±— ) (±κ )

n m n

±κ (aκ )

κ κ

m

±κ a† ±κ

n

( ±κ |aκ | ±κ ) ;

= (5.19)

κ

consequently, |±κ is called a coherent state. The de¬nition (5.18) shows that |±κ

belongs to the single-mode subspace Hκ ‚ HF that is spanned by the number states

for the mode E κ .

½¾ Coherent states

The new physical interpretation is clearest for the radiation modes of a physical

cavity. In the momentum-space representation, the operator pκ is just multiplication

by the eigenvalue Pκ , and the expansion (2.99) shows that the electric ¬eld oper-

√

ator is a function of the pκ s, so that E (r) φ± (Pκ ) = Eκ V E κ (r) φ± (Pκ ) , where

√

Eκ = Pκ / √ is the electric ¬eld strength associated with Pκ . The dimension-

0V

less function V E κ (r) is of order unity and describes the shape of the mode func-

tion. The corresponding result in the coordinate representation is B (r) φ± (Qκ ) =

√ √

Bκ V Bκ (r) φ± (Qκ ) with Bκ = kκ Qκ / 0 V = µ0 /V ωκ Qκ . Eliminating Pκ in

favor of Eκ allows the Gaussian factor in φ± (P ) to be expressed as

(Eκ ’ µκ )2

0V 2

exp ’ (Eκ ’ µκ ) = exp ’ , (5.20)

4e2

2 ωκ κ

where eκ is the vacuum ¬‚uctuation strength de¬ned by eqn (2.188). Thus a coherent

state displays a Gaussian probability density in the electric ¬eld amplitude Eκ with

average µκ , and variance V (Eκ ) = 2e2 . Similarly the coordinate-space wave function

κ

is a Gaussian in Bκ with average βκ and variance 2b2 . The classical limit corresponds

κ

to |Eκ | eκ and |Bκ | bκ , which are both guaranteed by |±κ | 1. As an example,

15 ’1

(»κ ≈ 2 µm) and V = 1 cm , then the vacuum ¬‚uctuation

3

consider ωκ = 10 s

strength for the electric ¬eld is eκ 0.08 V/m.

The fact that ±κ is a phasor provides the useful pictorial representation shown in

Fig. 5.1. This is equivalent to a plot in the phase plane (Qκ , Pκ ). The result (5.17) for

the wave function and the phase plot Fig. 5.1 are expressed in terms of the excitation

of a single radiation oscillator in a physical cavity, but the idea of coherent states is not

restricted to this case. The annihilation operator a can refer to a cavity mode (aκ ), a

(box-quantized) plane wave (aks ), or a general wave packet operator (a [w]), as de¬ned

in Section 3.5.2, depending on the physical situation under study. In the interests of

simplicity, we will initially consider situations in which only one annihilation operator a

(one electromagnetic degree of freedom) is involved. This is su¬cient for a large variety

Im (±)

±

±0

Fig. 5.1 The coherent state (displaced ground

state) |±0 is pictured as an arrow joining the

origin to the point ±0 in the complex plane.

The quantum uncertainties of the ground state

(at the origin) and the displaced ground state Re (±)

are each represented by an error circle (quan-

tum fuzzball ).

½¿

Sources of coherent states

of applications, but the physical justi¬cation for isolating the single-mode subspace

associated with a is that coupling between modes is weak. This fact should always be

kept in mind, since a more complete calculation may involve taking the weak coupling

into account, e.g. when considering dissipative or nonlinear e¬ects.

5.1.3 Coherent states in the number-state basis

We now consider a single mode and represent |± by the number-state expansion

∞

|± = bn |n . (5.21)

n=0

According to eqn (2.78) the eigenvalue equation (5.14) can then be written as

∞ ∞

√

nbn |n ’ 1 = ± bn |n . (5.22)

n=0 n=0

Equating the coe¬cients of the number states yields the recursion relation, bn+1 =

√

√

±/ n + 1 bn , which has the solution bn = b0 ±n / n!. Thus each coe¬cient bn is a

function of the complex parameter ±, in agreement with the discussion at the beginning

of the chapter. The vacuum coe¬cient b0 is chosen to get a normalized state, with the

result ∞

±n

’|±|2 /2

√ |n .

|± = e (5.23)

n!

n=0

This construction works for any complex number ±, so the spectrum of the operator a

is the entire complex plane. A similar calculation for a† fails to ¬nd any normalizable

solutions; consequently, a† has neither eigenvalues nor eigenvectors.

The average number of photons for the state |± is n = ± a† a ± = |±| , and the

2

probability that n is the outcome of a measurement of the photon number is

nn

Pn = e’n , (5.24)

n!

which is a Poisson distribution. The variance in photon number is

2 2

V (N ) = ± N 2 ± ’ ± |N | ± = |±| = n . (5.25)

5.2 Sources of coherent states

Coherent states are de¬ned by minimizing quantum ¬‚uctuations in the electromagnetic

¬eld, but the light emitted by a real source will display ¬‚uctuations for two reasons.

The ¬rst is that vacuum ¬‚uctuations of the ¬eld are inescapable, even in the absence

of charged particles. The second is that quantum ¬‚uctuations of the charged particles

in the source will imprint themselves on the emitted light. This suggests that a source

for coherent states should have minimal quantum ¬‚uctuations, and further that the

forces exerted on the source by the emitted radiation”the quantum back action”

should be negligible. The ideal limiting case is a purely classical current, which is so

½ Coherent states

strong that the quantum back action can be ignored. In this situation the material

source is described by classical physics, while the light is described by quantum theory.

We will call this the hemiclassical approximation, to distinguish it from the familiar

semiclassical approximation. The linear dipole antenna shown in Fig. 5.2 provides a

concrete example of a classical source.

In free space, the classical far-¬eld solution for the dipole antenna is an expanding

spherical wave with amplitude depending on the angle between the dipole p and the

radius vector r extending from the antenna to the observation point. A receiver placed

at this point would detect a ¬eld that is locally approximated by a plane wave with

propagation vector k = (ω/c) r/r and polarization in the plane de¬ned by p and r.

Another interesting arrangement would be to place the antenna in a microwave cavity.

In this case, d and ω could be chosen so that only one of the cavity modes is excited.

In either case, what we want now is the answer to the following question: What is the

quantum nature of the radiation ¬eld produced by the antenna?

We will begin with a quantum treatment of the charges and introduce the classical

limit later. For weak ¬elds, the A2 -term in eqn (4.32) for the Hamiltonian and the

A-term in eqn (4.43) for the velocity operator can both be neglected. In this approx-

imation the current operator and the interaction Hamiltonian are respectively given

by

pν

j (r) = δ (r’rν ) qν (5.26)

mν

ν

and

Hint = ’ d3 r j (r) · A (r) . (5.27)

This approximation is convenient and adequate for our purposes, but it is not strictly

necessary. A more exact treatment is given in (Cohen-Tannoudji et al., 1989, Chap.

III).

For an antenna inside a cavity, the positive-frequency part of the A-¬eld is

aκ E κ (r) ,

A(+) (r) = (5.28)

2 0 ωκ

κ

z

H

d /2

y

Fig. 5.2 A center fed linear dipole antenna

x

excited at frequency ω. The antenna is short,

i.e. d » = 2πc/ω.

½

Sources of coherent states

and the box-normalized expansion for an antenna in free space is obtained by E κ (r) ’

√

eks eik·r / V . Using eqn (5.28) in the expressions for Hem and Hint produces

ω κ a† aκ ,

Hem = (5.29)

κ

κ

and

d3 r j (r) · E — (r) + HC .

a†

Hint = ’ (5.30)

κ κ

2 0 ωκ

κ

In the Heisenberg picture, with aκ ’ aκ (t) and j (r) ’ j (r, t), the equation of

motion for aκ (t) is

‚

aκ (t) = [aκ (t) , H]

i

‚t

d3 r j (r, t) · E — (r) .

= ωκ aκ (t) ’ (5.31)

κ

2 0 ωκ

κ

In an exact treatment these equations would have to be solved together with the

Heisenberg equations for the charges, but we will avoid this complication by assum-

ing that the antenna current is essentially classical. The quantum ¬‚uctuations in the

current are represented by the operator

δ j (r, t) = j (r, t) ’ J (r, t) , (5.32)

where the average current is

J (r, t) = Tr ρchg j (r, t) , (5.33)

and ρchg is the density operator for the charges in the absence of any photons. The

expectation value J (r, t) represents an external classical current, which is analogous

to the external, classical electromagnetic ¬eld in the semiclassical approximation. With

this notation, eqn (5.31) becomes

‚

d3 r J (r, t) · E — (r)

aκ (t) = ωκ aκ (t) ’

i κ

‚t 2 0 ωκ

κ

(5.34)

d3 r δ j (r, t) · E — (r) .

’ κ

2 0 ωκ

κ

In the hemiclassical approximation the quantum ¬‚uctuation operator δ j (r, t) is ne-

glected compared to J (r, t), so the approximate Heisenberg equation is

‚

d3 r J (r, t) · E — (r) .

aκ (t) = ωκ aκ (t) ’

i (5.35)

κ

‚t 2 0 ωκ

κ

This is equivalent to approximating the Schr¨dinger-picture interaction Hamiltonian

o

by

HJ (t) = ’ d3 r J (r, t) · A (r) , (5.36)

which represents the quantized ¬eld interacting with the classical current J (r, t).

½ Coherent states

The Heisenberg equation (5.35) is linear in the operators aκ (t), so the individual

modes are not coupled. We therefore restrict attention to a single mode and simplify

the notation by {aκ , ωκ , E κ } ’ {a, ω, E}. The linearity of eqn (5.35) also allows us to

simplify the problem further by considering a purely sinusoidal current with frequency

„¦,

J (r, t) = J (r) e’i„¦t + J — (r) ei„¦t . (5.37)

With these simpli¬cations in force, the equation for a (t) becomes

‚

a (t) = ωa (t) ’ W e’i„¦t ’ W ei„¦t ,

i (5.38)

‚t

where

1

d3 r J (r) · E — (r) ,

W=

2 ωκ

0

(5.39)

1

d3 r J — (r) · E — (r) .

W=

2 0 ωκ

For this linear di¬erential equation the operator character of a (t) is irrelevant, and

the solution is found by elementary methods to be

a (t) = ae’iωt + ± (t) , (5.40)

where the c-number function ± (t) is

(ω+„¦)

sin t

∆

’i(ω+„¦)t/2 sin 2t 2

+ iW e’i∆t/2

± (t) = iW e , (5.41)

∆ ω+„¦

2 2

and ∆ = ω ’ „¦ is the detuning of the radiation mode from the oscillation frequency of

the antenna current. The ¬rst term has a typical resonance structure which shows”

as one would expect”that radiation modes with frequencies close to the antenna

frequency are strongly excited. The frequencies ω and „¦ are positive by convention,

so the second term is always o¬ resonance, and can be neglected in practice.

The use of the Heisenberg picture has greatly simpli¬ed the solution of this problem,

but the meaning of the solution is perhaps more evident in the Schr¨dinger picture.

o

The question we set out to answer is the nature of the quantized ¬eld generated by

a classical current. Before the current is turned on there is no radiation, so in the

Schr¨dinger picture the initial state is the vacuum: |Ψ (0) = |0 . In the Heisenberg

o

picture this state is time independent, and eqn (5.40) implies that a (t) |0 = ± (t) |0 .

Transforming back to the Schr¨dinger picture, by using eqn (3.83) and the identi¬ca-

o

tion of the Heisenberg-picture state vector with the initial Schr¨dinger-picture state

o

vector, leads to

a |Ψ (t) = ± (t) |Ψ (t) , (5.42)

where |Ψ (t) = U (t) |Ψ (0) is the Schr¨dinger-picture state that evolves from the

o

vacuum under the in¬‚uence of the classical current. Thus the radiation ¬eld from

a classical current is described by a coherent state |± (t) , with the time-dependent

amplitude given by eqn (5.41). According to Section 5.2, the ¬eld generated by the

classical current is the ground state of an oscillator displaced by Q (t) ∝ Re ± (t) and

P (t) ∝ Im ± (t).

½

Experimental evidence for Poissonian statistics

5.3 Experimental evidence for Poissonian statistics

Experimental veri¬cation of the predicted properties of coherent states, e.g. the Pois-

sonian statistics of photon number, evidently depends on ¬nding a source that produces

coherent states. The ideal classical currents introduced for this purpose in Section 5.2

provide a very accurate description of sources operating in the radio and microwave

frequency bands, but”with the possible exception of free-electron lasers”devices of

this kind are not found in the laboratory as sources for light at optical wavelengths.

Despite this, the folklore of laser physics includes the ¬rmly held belief that the out-

put of a laser operated far above threshold is well approximated by a coherent state.

This claim has been criticized on theoretical grounds (Mølmer, 1997), but recent ex-

periments using the method of quantum tomography, explained in Chapter 17, have

provided strong empirical support for the physical reality of coherent states. This

subtle question is beyond the scope of our book, so we will content ourselves with a

simple plausibility argument supporting a coherent state model for the output of a

laser. This will be followed by a discussion of an experiment performed by Arecchi

(1965) to demonstrate the existence of Poissonian photon-counting statistics”which

are consistent with a coherent state”in the output of a laser operated well above

threshold.

5.3.1 Laser operation above threshold

What is the basis for the folk-belief that lasers produce coherent states, at least when

operated far above threshold? A plausible answer is that the assumption of essentially

classical laser light is consistent with the mechanism that produces this light. The

argument begins with the assumption that, in the correspondence-principle limit of

high laser power, the laser ¬eld has a well-de¬ned phase. The phases of the individual

atomic dipole moments driven by this ¬eld will then be locked to the laser phase, so

that they all emit coherently into the laser ¬eld. The resulting reinforcement between

the atoms and the ¬eld produces a mutually coherent phase. Moreover, the re¬‚ection

of the generated light from the mirrors de¬ning the resonant cavity induces a positive

feedback e¬ect which greatly sharpens the phase of the laser ¬eld. In this situation

vacuum ¬‚uctuations in the light”the quantum back action mentioned above”have

a negligible e¬ect on the atoms, and the polarization current density operator ‚ P/‚t

behaves like a classical macroscopic quantity ‚P/‚t. Since ‚P/‚t oscillates at the

resonance frequency of the lasing transition, it plays the role of the classical current

in Section 5.2, and will therefore produce a coherent state.

The plausibility of this picture is enhanced by considering the operating conditions

in a real, continuous-wave (cw) laser. The net gain is the di¬erence between the gain

due to stimulated emission from the population of inverted atoms and the linear losses

in the laser (usually dominated by losses at the output mirrors). The increase of the

stimulated emission rate as the laser intensity grows causes depletion of the atomic

inversion; consequently, the gain decreases with increasing intensity. This phenomenon

is called saturation, and in combination with the linear losses it reduces the gain until

it exactly equals the linear loss in the cavity. This steady-state balance between the

saturated gain and the linear loss is called gain-clamping. Therefore, in the steady

state the intensity-dependent gain is clamped at a value exactly equal to the distributed

½ Coherent states

loss. The intensity of the light and the atomic polarization that produced it are in turn

clamped at ¬xed c-number values. In this way, the macroscopic atomic system becomes

insensitive to the quantum back-action of the radiation ¬eld, and acts like a classical

current source.

5.3.2 Arecchi™s experiment

In Fig. 5.3 we show a simpli¬ed description of Arecchi™s experiment, which measures

the statistics of photoelectrons generated by laser light transmitted through a ground-

glass disc. As a consequence of the transverse spatial coherence of the laser beam, light

transmitted through the randomly distributed irregularities in the disc will interfere

to produce the speckle pattern observed when an object is illuminated by laser light

(Milonni and Eberly, 1988, Sec. 15.8). In the far ¬eld of the disc, the transmitted light

passes through a pinhole”which is smaller than the characteristic spot size of the

speckle pattern”and is detected by a photomultiplier tube, whose output pulses enter

a pulse-height analyzer.

When the ground-glass disc is at rest, the light passing through the pinhole repre-

sents a single element of the speckle pattern.1 In this situation the temporal coherence

of the transmitted light is the same as that of original laser light, so the expectation

is that the detected light will be represented by a coherent state. Thus the photon

statistics should be Poissonian.

If the disc rotates so rapidly that the speckle features cross the pinhole in a time

short compared to the integration time of the detector, the transmitted light becomes

e¬ectively incoherent. As a simple classical model of this e¬ect, consider the vectorial

addition of phasors with random lengths (intensities) and directions (phases). The

resultant phasor is the solution to the 2D random-walk problem on the phasor plane.

In the limit of a large number of scatterers the distribution function for the resultant

phasor is a Gaussian centered at the origin. The incoherent light produced in this

way is indistinguishable from thermal light that has passed through a narrow spectral

¬lter. Therefore, one expects the resulting photon statistics to be described by the

Bose“Einstein distribution given by eqn (2.178).

Fig. 5.3 Schematic of Arecchi™s photon“

counting experiment. Light generated by a cw,

Helium’

helium“neon laser is transmitted through a

Photo-

neon

ground-glass disc to a small pinhole located

multiplier

laser

in the far ¬eld of the disc and placed in front

Rotating

of a photomultiplier tube. The resulting pho-

ground-

toelectron current is analyzed by means of a

glass disc Pulse-

pulse-height analyzer. Results for coherent (in-

height

coherent) light are obtained when the disc is

analyzer

stationary (rotating).

1 Murphy™s law dictates that the pattern element covering the pinhole will sometimes be a null in

the interference pattern. In practice the disc should be rotated until the signal is a maximum.

½

Experimental evidence for Poissonian statistics

Photomultiplier tubes are fast detectors, with nanosecond-scale resolution times, so

the pulse height (i.e. the peak voltage) of each output pulse is directly proportional to

the number of photons in the beam during a resolution time. This follows from the fact

that the fundamental detection process is the photoelectric e¬ect, in which (ideally)

a single photon would be converted to a single photoelectron. Thus two arriving pho-

tons would be converted at the photocathode into two photoelectrons, and so on. In

practice, due to the ¬nite thickness of the photocathode ¬lm, not all photons are con-

verted into photoelectrons. The fraction of photons converted to photoelectrons, which

is called the quantum e¬ciency, is studied in Section 9.1.3. Under the assumption that

the quantum e¬ciency is independent of the intensity of the light, and that the postde-

tection ampli¬cation system is linear, it is possible to convert the photoelectron-count

distribution, i.e. the pulse-height distribution, into the photon-count distribution func-

tion, p(n). In the ideal case when the quantum e¬ciency is 100%, each photon would

be converted into a photoelectron, and the photoelectron count distribution function

would be a faithful representation of p(n). However, it turns out that even if the quan-

tum e¬ciency is less than 100%, the photoelectron count distribution function will,

under these experimental conditions, still be a faithful representation of p(n).

In Fig. 5.4 the channel numbers on the horizontal axis label increasing pulse heights,

and the vertical coordinate of a point on the curve represents the number of pulses

counted within a small range (a bin) around the corresponding pulse height. One can

therefore view this plot as a histogram of the number of photoelectrons released in a

given primary event. The data points were obtained by passing the output pulse of

the photomultiplier directly into the pulse-height analyzer. This is raw data, in the

sense that the photomultiplier pulses have not been reshaped to produce standardized

digital pulses before they are counted. This avoids the dead-time problem, in which

the electronics cannot respond to a second pulse which follows too quickly after the

¬rst one.

Assuming that the photomultiplier (including its electron-multiplication struc-

tures) is a linear electronic system with a ¬xed integration time”given by an RC

time constant on the order of nanoseconds”the resulting pulse-height analysis yields

a faithful representation of the initial photoelectron distribution at the photocathode,

and hence of the photon distribution p(n) arriving at the photomultiplier. Therefore,

the channel number (the horizontal axis) is directly proportional to the photon number

n, while the number of counts (the vertical axis) is linearly related to the probability

p(n). For the case denoted by L (for laser light), the observed photoelectron distribu-

tion function ¬ts a Poissonian distribution, p(n) = exp (’n) nn /n!, to within a few per

cent. It is, therefore, an empirical fact that a helium“neon laser operating far above

threshold produces Poissonian photon statistics, which is what is expected from a co-

herent state. For the case denoted by G (for Gaussian light), the observed distribution

n+1

closely ¬ts the Bose“Einstein distribution p(n) = nn / (n + 1) , which is expected

for ¬ltered thermal light. The striking di¬erence between the nearly Poissonian curve

L and the nearly Gaussian curve G is the main result of Arecchi™s experiment.

Some remarks concerning this experiment are in order.

(1) As a function of time, the laser (with photon statistics described by the L-curve)

emits an ensemble of coherent states |± (t) , where ± (t) = |±|eiφ(t) . The amplitude

½¼ Coherent states

Number of

counts

107 . 8

G

L

7

6

5

4

3

2