Equivalent approaches to electromagnetic field quantization in a linear dielectric

0.24 MB·English

by  Fardin Kheirandish

Equivalent approaches to electromagnetic field
8 quantization in a linear dielectric
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n Fardin Kheirandish1 and Morteza Soltani1
∗ †
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J Department of Physics, University of Isfahan,

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Hezar Jarib Ave., Isfahan, Iran.
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] February 2, 2008
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Abstract
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[ It is shown that the minimal coupling method is equivalent to the

1 Huttner-Barnet and phenomenological approaches up to a canonical
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transformation.
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9 Keywords: Fieldquantization, DampedPolarizationModel,
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Phenomenological Model, Minimal Coupling Model, Linear
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1 Dielectric
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0 PACS number(s): 12.20.Ds
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1 Introduction
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a
One of the main developments in the quantum optics has been the study
of process, for example spontaneous emission, that take place in inside, or
adjacent to, material bodies. The need to interpret the experimental results
has stimulated attempts to quantize the electromagnetic field (EM) in a
material of general property [1].
∗fardin [email protected]
−
†[email protected]
1
In an inhomogeneous nondissipative medium, quantization have been
achieved by Glauber and Lewenstein [2]. But for dissipative medium, pres-
ence of absorbtion has the effect of coupling the EM field to a reservoir.
This matter is common in different quantization schemes [3, 4, 5, 6] since in
contrast to classical case, losses in quantum mechanics imply a coupling to
a reservoir whose degrees of freedom have to be added to the Hamiltonian.
But in different methods the coupling between reservoir and EM field has
been considered in different ways.
In damped polarization model the EM field is coupled with matter field
and matter field is coupled with a reservoir. This model is based on a La-
grangian and equal-time commutation relations (ETCR) can be written be-
tween canonical components. In this approach the Hamiltonian is obtained
from the Lagrangian and is diagonalized by the Fano technique [3].
In phenomenological method the reservoir act as a noise source and these
are conveniently represented by the Langevin force which act on the EM
field. The EM field is written in terms of the noise operators using the Green
function of Maxwell equations. The equivalence of these two methods has
been shown in different papers [4, 5, 6].
Recently a new method for dealing with quantum dissipative systems has
been introduced in [7, 8] and has been extended to EM field in a dissipative
media. This method is based on a Hamiltonian and the reservoir couples
directly to EM field. In this method the time revolution of Em field, in
contrast to damped polarization model, is obtained by solving the Heisen-
berg equations. The extension of this method to a magnetizable media is
straightforward [9].
The connection between this recent method and the other methods is the
subject of the present paper. Throughout the paper the SI units are used.
2 A brief review of different quantization meth-
ods
In this section we review the main approaches to EM field quantization in a
dielectric medium and then compare them.
2
2.1 The damped polarization model
This model is based on the Hopfield model of a dielectric [12]. The matter
is represented by a harmonic polarization field and a coupling between the
polarization field and another harmonic field is considered. Following the
standard approach in quantum electrodynamics, quantization procedure is
started from a Lagrangian density in real space
= + + + , (1)
em mat res int
L L L L L
where
ǫ 1
= 0E2 B2, (2)
em
L 2 − 2µ
0
is the electromagnetic part which can be expressed in terms of the vector
potential A and a scalar potential U ( B = A,E = A˙ U). The
∇ × − − ∇
Lagrangian of the matter (dielectric) is defined by
ρ ρω2
= X˙ 2 0X2, (3)
mat
L 2 − 2
where the field X is the polarization part, modeled by a harmonic oscillator
of frequency ω . The Lagrangian
0
= ∞dω(ρY˙ 2 ρω2Y2), (4)
Lres 2 ω − 2 ω
Z0
is the reservoir part, modeled by a set of harmonic oscillators used to model
the losses. The interaction part
∞
= α(A X˙ +U X) dωv(ω)X Y˙ , (5)
int ω
L − · ∇· − ·
Z0
includes the interaction between the light and the polarization field with
coupling constant α and the interaction between the polarization field and
theotheroscillatorswithafrequency dependent couplingv(ω). TakingX Y˙
ω
·
as the dissipating term is not essential but simplifies the calculations. The
displacement field D(r,t) is given by the following combination of electric
field and the material polarization
D(r,t) = ǫ E(r,t) αX(r,t). (6)
0
−
3
Since U˙ does not appear in the Lagrangian, U is not a proper dynamical
variable and the Lagrangian can be written in terms of the proper dynamical
variables A, X and Y . The easiest way to do this is to go to the reciprocal
ω
space and write all the fields in terms of spatial Fourier transforms. For
example the electric field can be written as
1
E(r,t) = d3kE˜(k,t)eik·r. (7)
3
(2π)2 Z
Since E(r,t) is a real field so E˜∗(k,t) = E˜( k,t). Therefore we can
−
restrict integration over k to half space. The total Lagrangian is
′
L = d3k(˜ + ˜ + ˜ + ˜ ), (8)
em mat res int
L L L L
Z
where the prime means that the integration is restricted to the half of the
reciprocal space and the Lagrangian densities in this space are defined by
˜ = ǫ (E˜2 c2B˜2), (9)
em 0
L −
˜ = ρX˜˙ 2 ρω2X˜2, (10)
Lmat − 0
˜ = ∞dω(ρY˜˙ 2 ρω2Y˜2), (11)
Lres ω − ω
Z0
˜ = α[A˜∗ X˜˙ +A˜ X˜˙ ∗ +ik (U˜∗X˜ U˜X˜∗)]
int
L − · · · −
∞dωv(ω)X˜∗ Y˜˙ +X˜ Y˜˙∗. (12)
− · ω · ω
Z0
Using theCoulomb gaugek A˜(k,t) = 0 andtheEuler-Lagrangeequation
·
for U˜˙∗, we find
α e (k) X˜(k,t)
U˜(k,t) = i ( 3 · ), (13)
ǫ k
0
where e (k) is the unit vector in the direction of k. The matter fields X˜ and
3
Y˜ can be decomposed into transverse and longitudinal parts. For example
ω
X˜ can be written as
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X˜(k,t) = X˜k(k,t)+X˜⊥(k,t), (14)
where k X˜⊥(k,t) = k X˜k(k,t) = 0 and similarly for Y˜ . The total
ω
· ×
Lagrangian can then be written as the sum of two independent parts. The
transverse part
′
L⊥ = d3k(˜⊥ + ˜⊥ + ˜⊥ + ˜⊥ ) (15)
Lem Lmat Lres Lint
Z
where
˜⊥ = ǫ (A˜˙ 2 c2k2A˜2), (16)
Lem 0 −
˜⊥ = (ρX˜˙ ⊥2 ρω2X˜⊥2), (17)
Lmat − 0
˜⊥ = ∞dω(ρY˜˙ ⊥2 ρω2Y˜⊥2), (18)
Lres ω − ω
Z0
˜⊥ = (αA˜ X˜˙ ⊥∗ + ∞dωv(ω)X˜⊥∗ Y˜˙ ⊥ +c.c.), (19)
Lint − · · ω
Z0
and the longitudinal part
′
Lk = d3k ˜k, (20)
L
Z
where
˜k = (ρX˜˙ k2 ρω2X˜k2 + ∞dωρY˜˙ k2 ρω2Y˜k2)
L − L ω − ω
Z0
∞
dω(v(ω)X˜k∗ Y˜k +c.c.) (21)
− · ω
Z0
In (21), ω = ω2 +ω2, is the longitudinal frequency and ω2 = α2. The
L 0 c c ρǫ0
link between theseqtwo parts is given by the total electric field
E˜(k,t) = E˜⊥(k,t)+E˜k(k,t) = A˜˙ (k,t)+ αX˜ke (k). (22)
3
− ǫ
0
Using (13) and the definition of the displacement field D given in (6), we
recover the fact that D(r,t) is a purely transverse field as expected.
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The vector potential in reciprocal space can be expanded in terms of the
unit vectors e (k) as
λ
A˜(k,t) = A˜ (k,t)e (k), (23)
λ λ
λ=1,2
X
where e (k) e′ (k) = δ and e (k) e(k) = 0 for λ = 1,2.
λ · λ λλ′ λ ·
TheLagrangian cannowbeusedtoobtainthecomponentsofconjugate
L
variables
ǫ E˜ = ∂L = ǫ A˜˙ , (24)
− 0 λ ∂A˜˙∗ 0 λ
λ
P˜ = ∂L = ρX˜˙ αA˜ , (25)
λ ∂X˜˙∗ λ − λ
λ
Q˜ = ∂£ = ρY˜˙ v(ω)Y˜ . (26)
ωλ ∂Y˜˙ ∗ ωλ − ωλ
ωλ
Using the Lagrangian(15) and the expressions for the conjugate variables
(24)-(26), we obtain the transverse part of Hamiltonian as
′
H⊥ = d3k( ˜⊥ + ˜⊥ + ˜⊥ ), (27)
Hem Hmat Hint
Z
where
˜⊥ = ǫ (E˜)⊥2 +ǫ c2k˜2A˜2, (28)
Hem 0 0
is the energy density of the EM field, k˜ is defined by k˜ = k2 +k2 with
c
k ωc = α2 , and q
c ≡ c ρc2ǫ0
q
P˜⊥2 ∞ Q˜⊥2
˜⊥ = +ρω˜2X˜⊥2+ dω( ω +ρω2Y˜⊥2)
Hmat ρ 0 ρ ω
Z0
+ ∞dω(ν(ω)X˜⊥∗ Q˜⊥)+c.c., (29)
ρ · ω
Z0
is the energy density of the matter field which includes the interaction be-
tween the magnetization and the reservoir. The frequency ω˜2 ω2 +
0 ≡ 0
∞dωv(ω)2 is the renormalized frequency of the polarization field and
0 ρ2
R
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α
H⊥ = d3k[A∗ P˜ +c.c.], (30)
int ρ ·
Z
is the interaction between the EM field and the magnetization.
By the same method we can obtain the longitudinal part of the Hamilto-
nian as
ˆ˜k = [Pˆ˜k2 +ρω˜2Xˆ˜k2+ ∞dω(Qˆ˜kω2 +ρω2Yˆ˜k2)
H ρ 0 ρ ω
Z0
+ ∞dω(ν(ω)Xˆ˜k∗ Qˆ˜k)+c.c.]+[α2Xˆ˜k2], (31)
ρ · ω ǫ
Z0 0
where we have separated theelectric part oftheHamiltonian fromthematter
part for later convenience.
The fields are quantized in a standard fashion by demanding ETCR be-
tween the variables and their conjugates. For EM field components we have
~
[Aˆ˜ (k,t),Eˆ˜∗ (k′,t)] = i δ δ(k k′), (32)
λ λ′ − ǫ λλ′ −
0
and for the matter
[Xˆ˜ (k,t),Pˆ˜∗(k′,t] = i~δ δ(k k′), (33)
λ λ′ λ,λ′ −
[Yˆ˜ (k,t),Qˆ˜∗ (k′,t)] = i~δ δ(k k′)δ(ω ω′), (34)
ωλ ω′λ′ λλ′ − −
with all other equal-time commutators being zero. Indeed, it is easily shown
thattheHeisenbergequationsofevolutionbasedontheHamiltonian(27)and
ETCR, (32)-(34), are identical to Maxwell equations in a dielectric medium
with displacement operator D(r,t), defined in (6).
The equations (27) and (6), together with the commutation relations in
(32) (34) complete the quantization procedure of transverse fields.
−
In order to extract useful information about the system, we need to solve
these equation. To facilitate the calculations, we introduce two sets of anni-
hilation operators
aˆ (k,t) = ǫ0 (k˜cAˆ˜ (k,t)+iEˆ˜ (k,t)), (35)
λ s2~k˜c λ λ
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ˆb (k,t) = ρ (ω˜ Xˆ˜ (k,t)+ iPˆ˜ (k,t)), (36)
λ 2~ω˜ 0 λ ρ λ
s 0
ˆb (k,t) = ρ ( iωYˆ˜ (k,t)+ 1Qˆ˜ (k,t)), (37)
ωλ 2~ω − ωλ ρ ωλ
r
where ω˜ is defined in(29). The different definitions forˆbandˆb onlyamount
0 ω
to a change of phase and have been chosen for further simplicity. From the
ETCR for the fields (32) (34) we obtain
−
[aˆ (k,t),aˆ†(k′,t)] = δ δ(k k′), (38)
λ λ λλ′ −
[ˆb (k,t),ˆb†(k′,t)] = δ δ(k k′), (39)
λ λ λλ′ −
[ˆb (k,t),ˆb† (kp,t)] = δ δ(k k′)δ(ω ω′). (40)
ωλ ω′λ′ λλ′ − −
We emphasize that, in contrast to the previous ETCR between the con-
jugate fields (32)-(34), which were correct only in half k space, the relations
(38) (40) are valid in the hole reciprocal space. Inverting eq (35) (37) to
− −
express the field operators in terms of the creation and annihilation oper-
ators and inserting them into the matter part of the Hamiltonian (27), we
obtain the normal ordered Hamiltonian for the transverse part of the matter
Hamiltonian as
Hˆ⊥ = Hˆ⊥ +Hˆ⊥ +Hˆ⊥ (41)
em matt int
∞
Hˆ⊥ = dω d3k ~ωaˆ†(ω,k)aˆ (ω,k), (42)
em λ λ
Z0 Z λ=1,2
X
∞
Hˆ⊥ = d3k [~ω˜ ˆb†(k,t)ˆb (k,t)+ dω~ωˆb† (k,t)ˆb (k,t)
mat 0 λ λ ωλ ωλ
o
Z λ=1,2 Z
X
~ ∞
+ dωV(ω)[ˆb†( k,t)+ˆb (k,t)][ˆb† ( k,t)+ˆb (k,t)]],
2 λ − λ ωλ − ωλ
Z0
(43)
~
Hˆ = i d3k Λ(k)(aˆ (k)+aˆ†( k))(ˆb (k) ˆb†( k)), (44)
int 2 λ λ − λ − λ −
Z λ=1,2
X
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where V(ω) [v(ω) ω ], Λ(k) ω˜0ckc2 and integration over k has been
≡ ρ ω˜0 ≡ k˜
extended to the wholeqreciprocal spaqce.
Now instead of solving the Heisenberg equations we use the Fano tech-
nique to diagonalize the Hamiltonian [10]. After diagonalization the field
operators are written in terms of the eigenoperators of the Hamiltonian.
The diagonalization is down in two step. In the first step the polarization
and reservoir part of the Hamiltonian is digitalized and the polarization is
written in terms of its eigenoperators. Then the total Hamiltonian is again
diagonalized with the same method.
In the first step the diagonalized expression for Hˆ⊥ is obtained as (the
mat
calculations leading to the diagonalization are lengthy and can be found in
[3] and we do not repeat them here)
∞
Hˆ⊥ = dω d3k ~ωBˆ†(ω,k)Bˆ (ω,k), (45)
matt λ λ
Z0 Z λ=1,2
X
where Bˆ†(k,ω) and Bˆ (k,ω) are the dressed matter field creation and anni-
λ λ
hilation operators and satisfy the usual ETCR
[Bˆ (ω,k,t),Bˆ† (ω′,k′,t)] = δ δ(k k′)δ(ω ω′). (46)
λ λ′ λλ′ − −
These operators can be expressed in terms of the initial creation and and
annihilation operators as
∞
B(k,ω) = α (ω)b(k)+β (ω)b†( k)+ dωα(ω,ω′)b(k,ω)+β(ω,ω′)b†( k,ω),
0 0
− −
Z0
(47)
and all the coefficient α (ω), β (ω), α (ω,ω′) and β (ω,ω′) can be obtained
0 0 1 1
in terms of microscopic parameters. In [3] the relation between α (ω) and
0
β (ω) is obtained as
0
ω ω˜
β (ω) = − 0α (ω), (48)
0 ω +ω˜ 0
0
and we will use this relation in the next section.
Using the commutators of ˆb with Bˆ and Bˆ† together with (47), we find
∞
ˆb (k) = dω[α∗(ω)Bˆ†(k,ω) β (ω)Bˆ ( k,ω)]. (49)
λ 0 λ − 0 λ −
Z0
The consistency of the diagonalization procedure is checked by verifying that
the initial commutation relation between ˆb(k) and ˆb†(k) are conserved.
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Using (44), (45), (49) and (41), the total Hamiltonian can be written as
∞
Hˆ = d3k{~ω˜kaˆ†λ(k)aˆλ(k)+ dω~ωBˆλ†(k,ω)Bˆλ(k,ω)
Z Z0
~ ∞
+ Λ(k) dω g(ω)Bˆ†(k,ω)[aˆ (k)+aˆ†( k)]+H.c. (50)
2 { λ λ λ − }
Z0
where Λ(k) is defined in (44) and g(ω) = iα (ω)+iβ (ω).
0 0
Now we should take the second step and diagonlaize the total Hamilto-
nian, (50), as
∞
Hˆ = d3k ~ω[Cˆ†(k.ω)Cˆ(k.ω) (51)
Z Z0
where eigenoperators of the system can be written as
Cˆ(k,ω) = α˜ (k,ω)aˆ(k)+β˜ (k,ω)aˆ†( k)
0 0
−
∞
+ dω′α˜(k,ω,ω′)Bˆ(k,ω,ω′)+β˜(k,ω,ω′)Bˆ†( k,ω′). (52)
−
Z0
To calculate the time dependence of EM field operators we should write
them in terms of eigenoperators of the system.
The vector potential is given by
1 ~
Aˆ(r,t) = d3k [aˆ (k,t)eik·r +H.c.]e (k). (53)
(2π)23 Z λ=1,2s2ǫ0ω˜k λ λ
X
By inverting (52), writing aˆ in terms of Cˆ and using the Hamiltonian
(51), to calculate the time dependence of Cˆ, we obtain aˆ(t) as
∞
aˆ (k,t) = dω[α˜∗(k,ω)Cˆ (k,ω)e−iωt β˜ (k,ω)Cˆ†(k,ω)eiωt]. (54)
λ 0 λ − 0 λ
Z0
Using (53) and (54) we obtain
~ ∞ ω Imχ(ω)
Aˆ(r,t) = ( )21 d3k dω
8π4ǫ ω2 +qω2(1+χ(ω))
0 Z Z0 k
[ Cˆ (k,ω)ei(k·r−ωt)e (k)+H.c.]. (55)
λ λ
×
λ=1,2
X
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