Field quantization in passive linear media

3.2.3 Positive- and negative-frequency parts

We are now in a position to explain the terms positive-frequency part and negative-

frequency part introduced in Section 3.1.2. For this purpose it is useful to review some

features of Fourier transforms. For any real function F (t), the Fourier transform sat-

is¬es F — (ω) = F (’ω). Thus F (ω) for negative frequencies is completely determined

by F (ω) for positive frequencies. Let us use this fact to rewrite the inverse transform

as ∞

dω

F (ω) e’iωt = F (+) (t) + F (’) (t) ,

F (t) = (3.97)

’∞ 2π

where the positive-frequency part,

∞

dω

F (ω) e’iωt ,

(+)

F (t) = (3.98)

2π

0

and the negative-frequency part,

0

dω

F (ω) e’iωt ,

(’)

F (t) = (3.99)

2π

’∞

are related by

F (’) (t) = F (+)— (t) . (3.100)

The de¬nitions of F (±) (t) guarantee that F (+) (ω) vanishes for ω < 0 and F (’) (ω)

vanishes for ω > 0.

The division into positive- and negative-frequency parts works equally well for any

time-dependent hermitian operator, X (t). One simply replaces complex conjugation

by the adjoint operation; i.e. eqn (3.100) becomes X (’) (t) = X (+)† (t). In particular,

the temporal Fourier transform of the operator A(+) (r, t), de¬ned by eqn (3.96), is

aks eks eik·r 2πδ (ω ’ ωk ) . (3.101)

A(+) (r, ω) = dt eiωt A(+) (r, t) =

2 0 ωk V

ks

Since ωk = c |k| > 0, A(+) (r, ω) vanishes for ω < 0, and A(’) (r, ω) = A(+)† (r, ’ω)

vanishes for ω > 0. Thus the Schr¨dinger-picture de¬nition (3.68) of the positive-

o

frequency part agrees with the Heisenberg-picture de¬nition at t = 0.

The commutation rules derived in Section 3.1.2 are valid here for equal-time com-

mutators, but for free ¬elds we also have the unequal-times commutators:

F (±) (r, t) , G(±) (r , t ) = 0 , (3.102)

provided only that F (±) (r, 0) and G(±) (r , 0) are sums over annihilation (creation)

operators.

3.3 Field quantization in passive linear media

Optical devices such as lenses, mirrors, prisms, beam splitters, etc. are the main tools

of experimental optics. In classical optics these devices are characterized by their bulk

Field quantization

optical properties, such as the index of refraction. In order to apply the same simple

descriptions to quantum optics, we need to extend the theory of photon propagation

in vacuum to propagation in dielectrics. We begin by considering classical ¬elds in

passive, linear dielectrics”which we will always assume are nonmagnetic”and then

present a phenomenological model for quantization.

3.3.1 Classical ¬elds in linear dielectrics

A review of the electromagnetic properties of linear media can be found in Appen-

dix B.5.1, but for the present discussion we only need to recall that the constitutive

relations for a nonmagnetic, dielectric medium are H (r, t) = B (r, t) /µ0 and

D (r, t) = 0E (r, t) + P (r, t) . (3.103)

For an isotropic, homogeneous medium that does not exhibit spatial dispersion (see

Appendix B.5.1) the polarization P (r, t) is related to the ¬eld by

dt χ(1) (t ’ t ) E (r, t ) ,

P (r, t) = (3.104)

0

where the linear susceptibility χ(1) (t ’ t ) describes the delayed response of the

medium to an applied electric ¬eld. Fourier transforming eqn (3.104) with respect to

time produces the equivalent frequency-domain relation

P (r, ω) = (ω) E (r, ω) .

(1)

0χ (3.105)

Applying the de¬nition of positive- and negative-frequency parts, given by eqns

(3.97)“(3.99), to the real classical ¬eld E (r, t) leads to

E (r, t) = E (+) (r, t) + E (’) (r, t) . (3.106)

In position space, the strength of the electric ¬eld at frequency ω is represented by

2

the power spectrum E (+) (r, ω) (see Appendix B.2). In reciprocal space, the power

2

spectrum is E (+) (k, ω) . We will often be concerned with ¬elds for which the power

spectrum has a single well-de¬ned peak at a carrier frequency ω = ω0 . The value

of ω0 is set by the experimental situation, e.g. ω0 is often the frequency of an injected

2

signal. The reality condition (3.100) for E (±) (r, ω) tells us that E (’) (r, ω) has a

peak at ω = ’ω0 ; consequently, the complete transform E (r, ω) has two peaks: one

at ω = ω0 and the other at ω = ’ω0 .

We will say that the ¬eld is monochromatic if the spectral width, ∆ω0 , of the

peak at ω = ±ω0 satis¬es

∆ω0 ω0 . (3.107)

We should point out that this usage is unconventional. Fields satisfying eqn (3.107)

are often called quasimonochromatic in order to distinguish them from the ideal case

in which the spectral width is exactly zero: ∆ω0 = 0. Since the ¬elds generated in real

experiments are always described by wave packets with nonzero spectral widths, we

prefer the de¬nition associated with eqn (3.107). The ideal ¬elds with ∆ω0 = 0 will

be called strictly monochromatic.

Field quantization in passive linear media

The concentration of the Fourier transform in the vicinity of ω = ±ω0 allows us to

(±)

de¬ne the slowly-varying envelope ¬elds E (r, t) by setting

(±)

(r, t) = E (±) (r, t) e±iω0 t ,

E (3.108)

so that

(+) (’)

(r, t) e’iω0 t + E

E (r, t) = E (r, t) eiω0 t . (3.109)

(’) (+)—

The slowly-varying envelopes satisfy E (r, t) = E (r, t), and the time-domain

version of eqn (3.107) is

(±) (±)

‚2E ‚E

(r, t) (r, t) (±)

ω0 E

2

ω0 (r, t) . (3.110)

‚t2 ‚t

The frequency-domain versions of eqns (3.108) and (3.109) are

(±)

E (r, ω) = E (±) (r, ω ± ω0 ) (3.111)

and

(+) (’)

E (r, ω) = E (r, ω ’ ω0 ) + E (r, ω + ω0 ) , (3.112)

(±)

respectively. The condition (3.107) implies that E (r, ω) is sharply peaked at ω = 0.

The Fourier transform of the vector potential is also concentrated in the vicinity

of ω = ±ω0 , so the slowly-varying envelope,

(+)

A (r, t) = A(+) (r, t) eiω0 t , (3.113)

satis¬es the same conditions. Since E (r, t) = ’‚A (r, t) /‚t, the two envelope functions

are related by

‚ (+)

(+) (+)

E (r, t) = iω0 A ’A. (3.114)

‚t

Applying eqn (3.110) to the vector potential shows that the second term on the right

side is small compared to the ¬rst, so that

(+) (+)

E (r, t) ≈ iω0 A . (3.115)

This is an example of the slowly-varying envelope approximation.

More generally, it is necessary to consider polychromatic ¬elds, i.e. superposi-

tions of monochromatic ¬elds with carrier frequencies ωβ (β = 0, 1, 2, . . .). The car-

rier frequencies are required to be distinct; that is, the power spectrum for a poly-

chromatic ¬eld exhibits a set of clearly resolved peaks at the carrier frequencies

ωβ . The explicit condition is that the minimum spacing between peaks, δωmin =

min [|ω± ’ ωβ | , ± = β] , is large compared to the maximum spectral width, ∆ωmax =

max [∆ωβ ]. The values of the carrier frequencies are set by the experimental situation

under study. The collection {ωβ } will generally contain the frequencies of any injected

¬elds together with the frequencies of radiation emitted by the medium in response to

¼ Field quantization

the injected signals. For a polychromatic ¬eld, eqns (3.108), (3.113), and (3.115) are

replaced by

(+)

(r, t) e’iωβ t ,

E (+) (r, t) = Eβ (3.116)

β

(+)

Aβ (r, t) e’iωβ t ,

A(+) (r, t) = (3.117)

β

and

(+) (+)

Eβ (r, t) = iωβ Aβ (r, t) . (3.118)

In the frequency domain, the total polychromatic ¬eld is given by

(σ)

E (r, ω) = E β (r, ω ’ σωβ ) , (3.119)

β σ=±

(±)

where each of the functions E β (r, ω) is sharply peaked at ω = 0.

A Passive, linear dielectric

An optical medium is said to be passive and linear if the following conditions are

satis¬ed.

(a) O¬ resonance. The classical power spectrum is negligible at frequencies that are

resonant with any transition of the constituent atoms. This justi¬es the assump-

tion that there is no absorption.

(b) Coarse graining. There are many atoms in the volume »3 , where »0 is the mean

0

wavelength for the incident ¬eld.

(c) Weak ¬eld. The ¬eld is not strong enough to induce signi¬cant changes in the

material medium.

(d) Weak dispersion. The frequency-dependent susceptibility χ(1) (ω) is essentially

constant across any frequency interval ∆ω ω.

(e) Stationary medium. The medium is stationary, i.e. the optical properties do

not change in time.

The passive property is incorporated in the o¬-resonance assumption (a) which

allows us to neglect absorption, stimulated emission, and spontaneous emission. The

description of the medium by the usual macroscopic coe¬cients such as the suscep-

tibility, the refractive index, and the conductivity is justi¬ed by the coarse-graining

assumption (b). The weak-¬eld assumption (c) guarantees that the macroscopic ver-

sion of Maxwell™s equations is linear in the ¬elds. The weak dispersion condition (d)

assures us that an input wave packet with a sharply de¬ned carrier frequency will

retain the same frequency after propagation through the medium. The assumption (e)

implies that the susceptibility χ(1) (t ’ t ) only depends on the time di¬erence t ’ t .

½

Field quantization in passive linear media

For later use it is helpful to explain these conditions in more detail. The medium

is said to be weakly dispersive (in the vicinity of the carrier frequency ω = ω0 ) if

‚χ(1) (ω)

χ(1) (ω0 )

∆ω0 (3.120)

‚ω ω=ω0

for any frequency interval ∆ω0 ω0 . We next recall that in a linear, isotropic dielectric

the vacuum dispersion relation ω = ck is replaced by

ωn (ω) = ck , (3.121)

where the index of refraction is related to the dielectric permittivity, (ω), by n2 (ω) =

(ω). Since (ω) can be complex”the imaginary part describes absorption or gain

(Jackson, 1999, Chap. 7)”the dispersion relation does not always have a real solu-

tion. However, for transparent dielectrics there is a range of frequencies in which the

imaginary part of the index is negligible.

For a given wavenumber k, let ωk be the mode frequency obtained by solving

the nonlinear dispersion relation (3.121), then the medium is transparent at ωk if

nk = n (ωk ) is real. In the frequency“wavenumber domain the electric ¬eld satis¬es

ω2 2

n (ω) ’ k 2 E k (ω) = 0 (3.122)

2

c

(see Appendix B.5.2, eqn (B.123)), so one ¬nds the general space“time solution

E (r, t) = E (+) (r, t) + E (’) (r, t), with

1

E (+) (r, t) = √ Eks eks ei(k·r’ωk t) . (3.123)

V ks

For a monochromatic ¬eld, the slowly-varying envelope is

1

(+)

(r, t) = √

E Eks eks ei(k·r’∆k t) , (3.124)

V ks

where the prime on the k-sum indicates that it is restricted to k-values such that

the detuning, ∆k = ωk ’ ω0 , satis¬es |∆k | ω0 . The wavelength mentioned in the

coarse-graining assumption (b) is then »0 = 2πc/ (n (ω0 ) ω0 ).

For a polychromatic ¬eld, eqn (3.108) is replaced by

(+)

(r, t) e’iωβ t ,

E (+) (r, t) = Eβ (3.125)

β

where

1

(+)

E β (r, t) = √ Eβks eks ei(k·r’∆βk t) , ∆βk = ωk ’ ωβ (3.126)

V ks

is a slowly-varying envelope ¬eld. The spectral width of the βth monochromatic ¬eld

(+) (+)

2 2

is de¬ned by the power spectrum E β (r, ω) or E β (k, ω) . The weak disper-

sion condition (d) is extended to this case by imposing eqn (3.120) on each of the

monochromatic ¬elds.

¾ Field quantization

The condition (3.107) for a monochromatic ¬eld guarantees the existence of an

intermediate time scale T satisfying

1 1

T , (3.127)

ω0 ∆ω0

i.e. T is long compared to the carrier period but short compared to the characteristic

time scale on which the envelope ¬eld changes. Averaging over the interval T will

wash out all the fast variations”on the optical frequency scale”but leave the slowly-

varying envelope unchanged. In the polychromatic case, applying eqn (3.107) to each

monochromatic component picks out an overall time scale T satisfying 1/ωmin T

1/∆ωmax , where ωmin = min (ωβ ).

B Electromagnetic energy in a dispersive dielectric

For an isotropic, nondispersive dielectric”e.g. the vacuum”Poynting™s theorem (see

Appendix B.5) takes the form

‚uem (r, t)

+ ∇ · S (r, t) = 0 , (3.128)

‚t

where

1 12

E 2 (r, t) + B (r, t)

uem (r, t) = (3.129)

2 µ0

is the electromagnetic energy density and

1

S (r, t) = E (r, t) — H (r, t) = E (r, t) — B (r, t) (3.130)

µ0

is the Poynting vector. The existence of an electromagnetic energy density is an es-

sential feature of the quantization schemes presented in Chapter 2 and in the present

chapter, so the existence of a similar object for weakly dispersive media is an important

question.

For a dispersive dielectric eqn (3.128) is replaced by

‚umag

pel (r, t) + +∇· S = 0, (3.131)

‚t

where the electric power density,

‚D (r, t)

pel (r, t) = E (r, t) · , (3.132)

‚t

is the power per unit volume ¬‚owing into the dielectric medium due to the action of

the slowly-varying electric ¬eld E, and

12

B (r, t)

umag (r, t) = (3.133)

2µ0

is the magnetic energy density; see Jackson (1999, Sec. 6.8). The existence of the

magnetic energy density umag (r, t) is guaranteed by the assumption that the material

¿

Field quantization in passive linear media

is not magnetically dispersive. The question is whether pel (r, t) can also be expressed

as the time derivative of an instantaneous energy density. The electric displacement

D (r, t) and the polarization P (r, t) are given by eqns (3.103) and (3.104), respectively,

so in general P (r, t) and D (r, t) depend on the electric ¬eld at times t = t. The

principle of causality restricts this dependence to earlier times, t < t, so that

χ(1) (t ’ t ) = 0 for t > t . (3.134)

(1)

For a nondispersive medium χ(1) (ω) has the constant value χ0 , so in this approxi-

mation one ¬nds that

(1)

χ(1) (t ’ t ) = χ0 δ (t ’ t ) . (3.135)

In this case, the polarization at a given time only depends on the ¬eld at the same

time. In the dispersive case, χ(1) (t ’ t ) decays to zero over a nonzero interval, 0 <

t’t < Tmem ; in other words, the polarization at t depends on the history of the electric

¬eld up to time t. Consequently, the power density pel (r, t) cannot be expressed as

pel (r, t) = ‚uel (r, t) /‚t, where uel (r, t) is an instantaneous energy density.

In the general case this obstacle is insurmountable, but for a monochromatic (or

polychromatic) ¬eld in a weakly dispersive dielectric it can be avoided by the use of

an appropriate approximation scheme (Jackson, 1999, Sec. 6.8). The fundamental idea

in this argument is to exploit the characteristic time T introduced in eqn (3.127) to

de¬ne the (running) time-average

T /2

1

pel pel (r, t + t ) dt .

(r, t) = (3.136)

T ’T /2

This procedure eliminates all rapidly varying terms, and one can show that

‚uel (r, t)

pel (r, t) = , (3.137)

‚t

where the e¬ective electric energy density is

d [ω0 (ω0 )] 1

E (r, t) · E (r, t)

uel (r, t) =

dω0 2

d [ω0 (ω0 )] (’) (+)

E (r, t) · E

= (r, t) , (3.138)

dω0

(+)

and E (r, t) is the slowly-varying envelope for the electric ¬eld. The e¬ective electric

energy density for a polychromatic ¬eld is a sum of terms like uel (r, t) evaluated

for each monochromatic component. We will use this expression in the quantization

technique described in Section 3.3.5.

3.3.2 Quantization in a dielectric

The behavior of the quantized electromagnetic ¬eld in a passive linear dielectric is

an important practical problem for quantum optics. In principle, this problem could

be approached through a microscopic theory of the quantized ¬eld interacting with

Field quantization

the point charges in the atoms constituting the medium. The same could be said for

the classical theory of ¬elds in a dielectric, but it is traditional”and a great deal

easier”to employ instead a phenomenological macroscopic approach which describes

the response of the medium by the linear susceptibility. The long history and great

utility of this phenomenological method have inspired a substantial body of work

aimed at devising a similar description for the quantized electromagnetic ¬eld in a

dielectric medium.1 This has proven to be a di¬cult and subtle task. The phenomeno-

logical quantum theory for the cavity and the exact vacuum theory both depend on

an expression for the classical energy as the sum of energies for independent radiation

oscillators, but”as we have seen in the previous section”there is no exact instanta-

neous energy for a dispersive medium. Fortunately, an exact quantization method is