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

1

Channel

number

0

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38

Fig. 5.4 Data from Arecchi™s experiment measuring photoelectron statistics of a cw, heli-

um“neon laser. The number of counts of output pulses from a photomultiplier tube, binned

within a narrow window of pulse heights, is plotted against the voltage pulse height for two

kinds of light ¬elds: ˜L™ for ˜laser light™, which closely ¬ts a Poissonian, and ˜G™ for ˜Gaussian

light™, which closely ¬ts a Bose“Einstein distribution function. (Reproduced from Arecchi

(1965).)

|± (t)| = |±| is ¬xed by gain clamping, but the phase φ(t) is not locked to any

external source. Consequently, the phase wanders (or di¬uses) on a very long

coherence time scale „coh 0.1 s (the inverse of the laser line width). The phase-

wander time scale is much longer than the integration time, RC 1 ns, of the

very fast photon detection system. Furthermore, the Poissonian distribution p(n)

√