stream of linearly-polarized photons, this process is described by saying that the light

™ ™

beam exerts a torque on the plate: „z = dSz /dt = N (’ ), where N is the rate of ¬‚ow

of photons through the plate. The resulting twist of the torsion ¬ber can be sensitively

measured by means of a small mirror attached to the ¬ber.

The original experiment actually used a steady stream of light composed of very

many photons, so a classical description would be entirely adequate. However, if the

sensitivity of the experiment were to be improved to a point where ¬‚uctuations in the

¼ Field quantization

angular position of the wave plate could be measured, then the discrete nature of the

angular momentum transfer of per photon to the wave plate would show up. The

transfer of angular momentum from an individual photon to the wave plate must in

principle be discontinuous in nature, and the twisting of the wave plate should manifest

a ¬ne, ratchet-like Brownian motion. The experiment to see such ¬‚uctuations”which

would be very di¬cult”has not been performed.

A more modern experiment to demonstrate the spin angular momentum of light

was performed by trapping a small, absorbing bead within the beam waist of a tightly

focused Gaussian laser beam (Friese et al., 1998). The procedure for trapping a small

particle inside the beam waist of a laser beam has been called an optical tweezer ,

since one can then move the particle around at will by displacing the axis of the light

beam. The accompanying procedure for producing arbitrary angular displacements of

a trapped particle by transferring controllable amounts of angular momentum from

the light to the particle has been called an optical torque wrench (Ashkin, 1980). For

linearly-polarized light, no e¬ect is observed, but switching the incident laser beam to

circular polarization causes the trapped bead to begin spinning around the axis de¬ned

by the direction of propagation of the light beam. In classical terms, this behavior is a

result of the torque exerted on the particle by the absorbed light. From the quantum

point of view absorption of each photon deposits of angular momentum in the bead;

therefore, the bead has to spin up in order to conserve angular momentum.

Observations of the orbital angular momentum, Lz , of light have also been made

using a similar technique (He et al., 1995). The experiment begins with a linearly-

polarized laser beam in a Gaussian TEM00 mode. This beam”which has zero helicity

and zero orbital angular momentum”then passes through a computer-generated holo-

graphic mask with a spiral pattern imprinted onto it. The linearly-polarized, paraxial,

Gaussian beam is thereby transformed into a linearly-polarized, paraxial Laguerre“

Gaussian beam of light (Siegman, 1986, Sec. 16.4). The output beam possesses orbital,

but no spin, angular momentum. A simple Laguerre“Gaussian mode is one in which

the light e¬ectively orbits around the axis of propagation as if in an optical vortex

with a given sense of circulation. The transverse intensity pro¬le is doughnut-shaped,

with a null at its center marking a phase singularity in the beam. In principle, the spi-

ral holographic mask would experience a torque resulting from the transfer of orbital

angular momentum”one unit (+ ) per photon”to the light beam from the mask.

However, this experiment has not been performed.

What has been observed is that a small, absorbing bead trapped at the beam waist

of a Laguerre“Gaussian mode”with nonzero orbital angular momentum”begins to

spin. This spinning motion is due to the steady transfer of orbital angular momentum

from the light beam into the bead by absorption. The resultant torque is given by

™ ™

„z = dLz /dt = N (’ ), where N is the rate of photon ¬‚ow through the bead. Again,

there is a completely classical description of this experiment, so the photon nature of

light need not be invoked.

Just as for the spin-transfer experiments, a su¬ciently sensitive version of this

experiment, using a small enough bead, would display the discontinuous transfer of

orbital angular momentum in the form of a ¬ne, ratchet-like Brownian motion in

the angular displacement of the bead. This would be analogous to the discontinuous

½

Field quantization in the vacuum

transfer of linear momentum due to impact of atoms on a pollen particle that results

in the random linear displacements of the particle seen in Brownian motion. This

experiment has also not been performed.

3.1.4 Box quantization

The local, position-space commutation relations (3.1) and (3.3)”or the equivalent

reciprocal-space versions (3.25) and (3.26)”do not require any idealized boundary

conditions, but the right sides of eqns (3.3) and (3.26) contain singular functions

that cause mathematical problems, e.g. the improper one-photon state |1ks . On the

other hand, the cavity mode operators aκ and a† ”which do depend on idealized

κ

boundary conditions”have discrete labels and the one-photon states |1κ = a† |0 are

κ

normalizable. As usual, we would prefer to have the best of both worlds; and this can

be accomplished”at least formally”by replacing the Fourier integral in (3.5) with

a Fourier series. This is done by pretending that all ¬elds are contained in a ¬nite

volume V , usually a cube of side L, and imposing periodic boundary conditions at the

walls, as explained in Appendix A.4.2. This is called box quantization. Since this

imaginary cavity is not de¬ned by material walls, the periodic boundary conditions

have no physical signi¬cance. Consequently, meaningful results are only obtained in

the limit of in¬nite volume. Thus box quantization is a mathematical trick; it is not a

physical idealization, as in the physical cavity problem.

The mathematical situation resulting from this trick is almost identical to

that of the ideal physical cavity. For this case, the traveling waves, fks (r) =

√

eks exp (ik · r) / V , play the role of the cavity modes. The periodic boundary condi-

tions impose k =2πn/L, where n is a vector with integer components. The f ks s are

an orthonormal set of modes, i.e.

—

d3 r fks (r) · fk s (r) = δkk δss .

(fks , fk s ) = (3.63)

V

The various expressions for the commutation relations, the ¬eld operators, and

the observables can be derived either by replacing the real cavity mode functions in

Chapter 2 by the complex modes f ks (r), or by applying the rules relating Fourier

integrals to Fourier series, i.e.

√

d3 k 1

” and as (k) ” V aks , (3.64)

3 V

(2π) k

to the expressions obtained in Sections 3.1.1“3.1.3. In either way, the commutation

relations and the number operator are given by

aks , a† s a† aks .

= δkk δss , [aks , ak s ] = 0 , N = (3.65)

k ks

ks

The number states are de¬ned just as for the physical cavity,

nks

a†

ks

√ |0 ,

|n = (3.66)

nks !

ks

¾ Field quantization

where n = {nks } is the set of occupation numbers, and the completeness relation is

|n n| = 1 . (3.67)

n

Thus the box-quantization scheme replaces the delta function in eqn (3.26) by the

ordinary Kronecker symbol in the discrete indices k and s. Consequently, the box-

quantized operators aks are as well behaved mathematically as the physical cavity

operators aκ . This allows the construction of the Fock space to be carried out in

parallel to Chapter 2.1.2-C.

The expansions for the ¬eld operators are

A(+) (r) = aks eks eik·r , (3.68)

2 0 ωk V

ks

ωk

E(+) (r) = aks eks eik·r ,

i (3.69)

2 0V

ks

and

k

B(+) (r) = saks eks eik·r , (3.70)

2 0 cV

ks

where the expansion for B(+) was obtained by using B = ∇ — A and the special

property (B.52) of the circular polarization basis.

The Hamiltonian, the momentum, and the helicity are respectively given by

ωk a† aks ,

Hem = (3.71)

ks

ks

ka† aks ,

P= (3.72)

ks

ks

and

ksa† aks .

S= (3.73)

ks

ks

As always, these achievements have a price. One part of this price is that physically

meaningful results are only obtained in the limit V ’ ∞. This is not a particularly

onerous requirement, since getting the correct limit is simply a matter of careful al-

gebra combined with the rules in eqn (3.64). A more serious issue is the absence of

the total angular momentum from the list of observables in eqns (3.71)“(3.73). One

way of understanding the problem here is that the expression (3.55) for L contains

the di¬erential operator ‚/‚k which creates di¬culties in converting the continuous

integral over k into a discrete sum. The alternative expression (3.57) for L does not

involve k, so it might seem to o¬er a solution. This hope also fails, since the r-integral

in this representation must now be carried out over the imaginary cube V . The edges

of the cube de¬ne preferred directions in space, so there is no satisfactory way to de¬ne

the orbital angular momentum L.

¿

The Heisenberg picture

3.2 The Heisenberg picture

The quantization rules in Chapter 2 and Section 3.1.1 are both expressed in the

Schr¨dinger picture: observables are represented by time-independent hermitian oper-

o

ators X (S) , and the state of the radiation ¬eld is described by a ket vector Ψ(S) (t) ,

obeying the Schr¨dinger equation

o

‚

Ψ(S) (t) = H (S) Ψ(S) (t) ,

i (3.74)

‚t

or by a density operator ρ(S) (t), obeying the quantum Liouville equation (2.119)

‚ (S)

ρ (t) = H (S) , ρ(S) (t) .

i (3.75)

‚t

The superscript (S) has been added in order to distinguish the Schr¨dinger picture

o

from two other descriptions that are frequently used. Note that the density operator is

an exception to the rule that Schr¨dinger-picture observables are independent of time.

o

There is an alternative description of quantum mechanics which actually preceded

the familiar Schr¨dinger picture. In Heisenberg™s original formulation”which appeared

o

one year before Schr¨dinger™s”there is no mention of a wave function or a wave equa-

o

tion; instead, the observables are represented by in¬nite matrices that evolve in time

according to a quantum version of Hamilton™s equations of classical mechanics. This

form of quantum theory is called the Heisenberg picture; the physical equivalence of

the two pictures was subsequently established by Schr¨dinger. The Heisenberg picture

o

is particularly useful in quantum optics, especially for the calculation of correlations

between measurements at di¬erent times. A third representation”called the interac-

tion picture”will be presented in Section 4.8. It will prove useful for the formulation

of time-dependent perturbation theory in Section 4.8.1. The interaction picture also

provides the foundation for the resonant wave approximation, which is introduced in

Section 11.1.

In the following sections we will study the properties of the Schr¨dinger and Heisen-

o

berg pictures and the relations between them. In order to distinguish between the same

quantities viewed in di¬erent pictures, the states and operators will be decorated with

superscripts (S) or (H) for the Schr¨dinger or Heisenberg pictures respectively. In

o

applications of these ideas the superscripts are usually dropped, and the distinctions

are”one hopes”made clear from context.

The Heisenberg picture is characterized by two features: (1) the states are inde-

pendent of time; (2) the observables depend on time. Imposing the superposition prin-

ciple on the Heisenberg picture implies that the relation between the time-dependent,

Schr¨dinger-picture state vector Ψ(S) (t) and the corresponding time-independent,

o

Heisenberg-picture state Ψ(H) must be linear. If we impose the convention that the

two pictures coincide at some time t = t0 , then there is a linear operator U (t ’ t0 )

such that

Ψ(S) (t) = U (t ’ t0 ) Ψ(H) . (3.76)

The identity of the pictures at t = t0 , Ψ(H) = Ψ(S) (t0 ) , is enforced by the initial

condition U (0) = 1. Substituting eqn (3.76) into the Schr¨dinger equation (3.74) yields

o

the di¬erential equation

Field quantization

‚

U (t ’ t0 ) = H (S) U (t ’ t0 ) , U (0) = 1

i (3.77)

‚t

for the operator U (t ’ t0 ). This has the solution (Bransden and Joachain, 1989, Sec.

5.7)

i

U (t ’ t0 ) = exp ’ (t ’ t0 ) H (S) , (3.78)

where the evolution operator on the right side is de¬ned by the power series for the

exponential, or by the general rules outlined in Appendix C.3.6. The Hermiticity of

H (S) guarantees that U (t ’ t0 ) is unitary, i.e.

U (t ’ t0 ) U † (t ’ t0 ) = U † (t ’ t0 ) U (t ’ t0 ) = 1 . (3.79)

The choice of t0 is dictated by convenience for the problem at hand. In most cases

it is conventional to set t0 = 0, but in scattering problems it is sometimes more useful

to consider the limit t0 ’ ’∞. The evolution operator satis¬es the group property,

U (t1 ’ t2 ) U (t2 ’ t3 ) = U (t1 ’ t3 ) , (3.80)

which simply states that evolution from t3 to t2 followed by evolution from t2 to t1 is

the same as evolving directly from t3 to t1 . For the special choice t0 = 0, this simpli¬es

to U (t1 ) U (t2 ) = U (t1 + t2 ). The de¬nition (3.78) also shows that U (’t) = U † (t).

In what follows, we will generally use the convention t0 = 0; any other choice of initial

time will be introduced explicitly.

The physical equivalence of the two pictures is enforced by requiring that each

Schr¨dinger-picture operator X (S) and the corresponding Heisenberg-picture operator

o

(H)

X (t) have the same expectation values in corresponding states:

Ψ(H) X (H) (t) Ψ(H) = Ψ(S) (t) X (S) Ψ(S) (t) , (3.81)

for all vectors Ψ(S) (t) and observables X (S) . Using eqn (3.76) allows this relation to

be written as

Ψ(H) X (H) (t) Ψ(H) = Ψ(H) U † (t) X (S) U (t) Ψ(H) . (3.82)

Since this equation holds for all states, the general result (C.15) shows that the oper-

ators in the two pictures are related by

X (H) (t) = U † (t) X (S) U (t) . (3.83)

Note that the Heisenberg-picture operators agree with the (time-independent)

Schr¨dinger-picture operators at t = 0. This de¬nition, together with the group prop-

o

erty U (t1 ) U (t2 ) = U (t1 + t2 ), provides a useful relation between the Heisenberg

operators at di¬erent times:

X (H) (t + „ ) = U † (t + „ ) X (S) U (t + „ )

= U † („ ) U † (t) X (S) U (t) U („ )

= U † („ ) X (H) (t) U („ ) . (3.84)

Also note that H (S) commutes with exp ±itH (S) / , so eqn (3.83) implies that the

Hamiltonian is the same in both pictures: H (H) (t) = H (S) = H.

The Heisenberg picture

In the Heisenberg picture, the operators evolve in time while the state vectors are

¬xed. The density operator is again an exception. Applying the transformation (3.83)

to the de¬nition of the Schr¨dinger-picture density operator,

o

Pu ˜(S) (t)

ρ(S) (t) = ˜(S) (t) , (3.85)

u u

u

yields the time-independent operator

Pu ˜(H)

ρ(H) = ˜(H) = ρ(S) (0) , (3.86)

u u

u

which is the initial value for the quantum Liouville equation (3.75).

A di¬erential equation describing the time evolution of operators in the Heisenberg

picture is obtained by combining eqn (3.77) with the common form of the Hamiltonian

to get

‚X (H) (t) i

= U † (t) H, X (S) U (t)

‚t

i

H, X (H) (t) ,

= (3.87)

where the last line follows from the identity

U † (t) X (S) Y (S) U (t) = U † (t) X (S) U (t) U † (t) Y (S) U (t)

= X (H) (t) Y (H) (t) . (3.88)

Multiplying eqn (3.87) by i yields the Heisenberg equation of motion for the

observable X (H) :

‚X (H) (t)

= X (H) (t) , H .

i (3.89)

‚t

The de¬nition (3.83) provides a solution for this equation. The name ˜constant of the

motion™ for operators X (S) that commute with the Hamiltonian is justi¬ed by the

observation that the Heisenberg equation for X (H) (t) is (‚/‚t) X (H) (t) = 0.

In most applications we will suppress the identifying superscripts (H) and (S). The

distinctions between the Heisenberg and Schr¨dinger pictures will be maintained by the

o

convention that an operator with a time argument, e.g. X (t), is the Heisenberg-picture

form, while X”with no time argument”signi¬es the Schr¨dinger-picture form. The

o

only real danger of this convention is that density operators behave in the opposite way;

ρ (t) denotes a Schr¨dinger-picture operator, while ρ is taken in the Heisenberg picture.

o

This is not a serious problem if the accompanying text provides the appropriate clues.

3.2.1 Equal-time commutators

A pair of Schr¨dinger-picture operators X and Y is said to be canonically conjugate

o

if [X, Y ] = β, where β is a c-number. Canonically conjugate pairs, e.g. position and

momentum, play an important role in quantum theory, so it is useful to consider the

commutator in the Heisenberg picture. Evaluating [X (t) , Y (t )] for t = t requires a

Field quantization

complete solution of the Heisenberg equations for X (t) and Y (t ), but the equal-time

commutator for such a canonically conjugate pair is given by

[X (t) , Y (t)] = U † (t) XU (t) , U † (t) Y U (t)

= U † (t) [X, Y ] U (t)

= β. (3.90)

Thus the equal-time commutator of the Heisenberg-picture operators is identical to the

commutator of the Schr¨dinger-picture operators. Applying this to the position-space

o

commutation relation (3.3) and to the canonical commutator (3.65) yields

i

∆⊥ (r ’ r )

[Ai (r, t) , ’Ej (r , t)] = (3.91)

ij

0

and

aks (t) , a† s (t) = δss δkk , (3.92)

k

respectively.

3.2.2 Heisenberg equations for the free ¬eld

The preceding arguments are valid for any form of the Hamiltonian, but the results are

particularly useful for free ¬elds. The Heisenberg-picture form of the box-quantized

Hamiltonian is

ωk a† (t) aks (t) ,

Hem = (3.93)

ks

ks

and eqn (3.89), together with the equal-time versions of eqn (3.65), yields the equation

of motion for the annihilation operators

d

’ ωk aks (t) = 0 .

i (3.94)

dt

The solution is

aks (t) = aks e’iωk t = eiHem t/ aks e’iHem t/ , (3.95)

where we have used the identi¬cation of aks (0) with the Schr¨dinger-picture operator

o

aks . Combining this solution with the expansion (3.68) gives

A(+) (r, t) = aks eks ei(k·r’ωk t) . (3.96)

2 0 ωk V

ks

The expansions (3.69) and (3.70) allow the operators E(+) (r, t) and B(+) (r, t) to be