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(’ ) of angular momentum in the opposite direction. In the classical limit of a steady
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

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