Canonical quantization of electromagnetic field in an anisotropic polarizable and magnetizable medium with spatial-temporal dispersion

0.19 MB·English

by  Majid Amooshahi

Canonical quantization of the electromagnetic
0 field in an anisotropic polarizable and
1
0
2 magnetizable medium

n
a
J M. Amooshahi1 ∗
9
2 1 Faculty of science, University of Isfahan ,Hezar Jarib Ave., Isfahan,Iran

]
h January 29, 2010
p
-
t
n
a
u
q Abstract
[
A fully canonical quantization of electromagnetic field is intro-
4
duced in the presence of an anisotropic polarizable and magnetizable
v
5 medium . Two tensor fields which couple the electromagnetic field
9
with the medium and have an important role in this quantization
6
2 method are introduced. The electric and magnetic polarization fields
. ofthemediumnaturallyareconcludedintermsofthecouplingtensors
0
1 and the dynamical variables modeling the magnetodielectric medium.
8 In Heisenberg picture, the constitutive equations of the medium to-
0
gether with the Maxwell laws are obtained as the equations of motion
:
v
of the total system and the susceptibility tensors of the medium are
i
X calculated in terms of the coupling tensors. Following a perturbation
r method the Green function related to the total system is found and
a
the time dependence of electromagnetic field operators is derived.
Keywords: Canonicalfieldquantization,Magnetodielectricmedium,
conductivity tensor, Susceptibility tensor, Coupling tensor, Constitu-
tive equation
PACS No: 12.20.Ds, 42.50.Nn
∗[email protected]
1
1 introduction
One of the most important quantum dissipative systems is the quantized
electromagnetic field in the presence of an absorbing polarizable medium. In
thiscasetherearemainlytwoquantizationapproaches, thephenomenological
method [1]-[7] and the damped polarization model[8, 9]. The phenomenolog-
ical scheme has been formulated on the basis of the fluctuation- dissipation
theorem [10]. In this method by adding a fluctuating noise term, that is the
noise polarization field, to the classical constitutive equation of the medium,
this equation is taken as the definition of the electric polarization operator.
Combination of the Maxwell equations and the constitutive equation in the
frequency domain, gives the electromagnetic field operators in terms of the
noise polarization field and the classical Green tensor. A set of bosonic op-
erators is associated with the noise polarization which their commutation
relations are given in agreement with the fluctuation- dissipation theorem.
This quantization scheme has been quite successful in describing some elec-
tromagneticphenomena inthepresence ofa lossy dielectric medium [11]-[15].
The phenomenological approach has been extended to a lossy magnetic or
anisotropic medium [16]-[18]. This formalism has also been generalized to an
arbitrary linearly responding medium based on a spatially nonlocal conduc-
tivity tensor [19].
The damped polarizationmodel to quantize electromagnetic field in a disper-
sive dielectric medium [8, 9] is a canonical quantization in which the electric
polarizationfield of themedium applies inthe Lagrangianof the total system
as a part of the degrees of freedom of the medium. The other parts of the
degrees of freedom of the absorbing medium are related to the dynamical
variables of a heat bath describing the absorptivity feature of the medium.
In this method the dielectric function of the medium is found in terms of
the coupling function of the heat bath and the polarization field, so that it
satisfies the Kramers-Kronig relations [20]. This quantization method has
been generalized to an inhomogeneous medium[21].
Inthepresent workwegeneralizeourpreviousmodel[22,23]toananisotropic
dispersive magnetodielectric medium using a canonical approach. In this
formalism the medium is modeled with two independent collections of vec-
tor fields. These collections solely constitute the degrees of freedom of the
medium and it is not needed the electric and magnetic polarization fields to
be included in the Lagrangian of the total system as a part of the degrees
of freedom of the medium as in the Huttner-Barnett model [8]. In fact the
2
dynamical fields modeling the dispersive medium are able to describe both
polarizability and absorptivity features of the medium.
This paper is organized as follows. In Sec. 2, a Lagrangian for the to-
tal system is proposed and classical electrodynamics in the presence of an
anisotropic polarizable and magnetizable medium with spatial-temporal dis-
persion briefly is discussed . In Sec. 3, applying the Lagrangian introduced
in Sec. 2 a fully canonical quantization of both electromagnetic field and
the dynamical variable modeling the responding medium is demonstrated.
Then in Sec. 4, the constitutive equations of the medium are obtained as the
consequences of the Heisenberg equations of the total system and the electric
and magnetic susceptibility tensors of the medium are calculated in terms of
the parameters applied in the theory . In Sec.5, it is shown that the Green
function of the total system in reciprocal space satisfies an algebraic equa-
tion and a perturbation method to obtain the Green function is introduced.
Finally in section 6 the model is modified for media which the distinction
between polarization and magnetization is not possible. This paper is closed
with a summary and some concluding remarks in Sec.7 .
2 Classical electrodynamics in an anisotropic
magnetodielectric medium
In order to present a fully canonical quantization of electromagnetic field
in the presence of an anisotropic polarizable and magnetizable medium, we
model the medium by two independent reservoirs. Each reservoir contains a
continium of three dimensional harmonic oscillators labeled with a continu-
ous parameter ω. We call these two continuous sets of oscillators ”E field ”
and ”M field”. The E field and M field describe polarizability and magne-
tizability of the medium, respectively. This means that, in this approach it
is not needed the electric and magnetic polarization fields of the medium to
be appeared explicitly in the Lagrangian of the total system as a part of the
degrees of freedom ofthe medium , but thecontribution of themedium inthe
Lagrangianof the total system is related only to the Lagrangianof the E and
M fields and these fields completely describe the degrees of freedom of the
medium. The presence of the ”E field” in the total Lagrangian is sufficient
for a complete description of both polarizability and the absorption of the
3
medium due to its electrically dispersive property. Also the ”M field” solely
is sufficient in order to description of both magnetizability and the absorp-
tion of the medium due to its magnetically dispersive property. Therefore,
in order to have a classical treatment of electrodynamics in a magnetodielec-
tric medium, we start with a Lagrangian for the total system (medium +
electromagnetic field ) which is the sum of three parts
L(t) = L +L +L (1)
res em int
where L is the part related to the degrees of freedom of the medium and
res
is the sum of the Lagrangians of the E and M fields
L = L +L (2)
res e m
where
L = ∞dω d3r 1X~˙ ·X~˙ − 1ω2X~ ·X~ (3)
e ω ω ω ω
2 2
Z0 Z (cid:20) (cid:21)
and
L = ∞dω d3r 1Y~˙ ·Y~˙ − 1ω2Y~ ·Y~ (4)
m ω ω ω ω
2 2
Z0 Z (cid:20) (cid:21)
Here the fields X~ and Y~ are the dynamical variables of the E and M fields,
ω ω
respectively .
In (1) L is the contribution of the electromagnetic field in the La-
em
grangian of the total system
1 B~2
L = d3r ε E~2 − (5)
em 0
2 2µ
" 0#
Z
and L is the part describing the interaction of the electromagnetic field
int
with the medium
∞
L = dω d3r d3r′f (ω,~r,r~′)Ei(~r,t)X~j(r~′,t)+
int ij ω
Z0 Z Z
∞
dω d3r d3r′g (ω,~r,r~′)Bi(~r,t)Y~j(r~′,t) (6)
ij ω
Z0 Z Z
The contributions L and L in the Lagrangian L are equivalent to the
e m res
consequences of diagonalization processes of the matter fields in the Huttner-
Barnet model [8]. That is, L ( L ) is equivalent to the diagonalization
e m
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of the contributions of related to three parts in the Huttner-Barnet model:
the dynamical variable describing the electric polarization (magnetic polar-
ization)of the medium , a heat-bath B ( B′) interacting with the electric
polarization (magnetic polarization ) and the interaction term between the
heat-bath B ( B′) and the electric polarization (magnetic polarization). In
the present approach modeling the medium , in a phenomenological way,
with two independent set of oscillators the lengthy diagonalization processes
have been eliminated in the start of this quantization scheme. Particularly
the diagonalization processes may be more tremendous for an anisotropic
medium.
In equations (5) and (6) E~ = −∂A~ − ∇~ϕ and B~ = ∇ × A~ are electric
∂t
and magnetic fields respectively, where A~ and ϕ are the vector and the scalar
potentials. The tensors f and g in (6) are called the coupling tensors of the
medium with electromagnetic field and for an inhomogeneous medium are
dependent on the both position vectors ~r and r~′. The coupling tensors are
the key parameters of this theory. As was mentioned above, it is not needed
the electric and magnetic polarization fields of the medium explicitly to be
appeared in the total Lagrangian (1)-(6) as a part of degrees of freedom of
the medium . As we will see, the electric polarization( magnetic polarization
) of the medium is obtained in terms of the coupling tensor f ( g ) and the
dynamical variables X~ ( Y~ ). Also the electric susceptibility tensor ( mag-
ω ω
netic susceptibility tensor) of the medium naturally will expressed in terms
of the coupling tensor f ( g ). The coupling tensors f and g are appeared as
common factors in both the noise polarization fields and the susceptibility
tensors of the medium, so that for the free space the susceptibility tensors
together with the noise polarizations become identically zero and this quanti-
zation scheme is reduced to the usual quantization of electromagnetic field in
free space. Furthermore when the medium tends to a non-absorbing one, the
coupling tensors and the noise polarizations tend also to zero and this quan-
tization method is reduced to the quantization in a non-absorbing medium
[22, 24].
In order to prevent some difficulties with a non-local Lagrangian such
as in (6), it is the easiest way to work in the reciprocal space and write all
the fields and the coupling tensors f , g in terms of their spatial Fourier
5
transforms. For example the dynamical variable X~ can be written as
ω
1
X~ (~r,t) = d3k X~ (~k,t) eı~k·~r (7)
ω ω
(2π)3)
Z
Since we are concerned with repal valued fields in the total Lagrangian(1)-(6),
~ ∗ ~ ~ ~ ~
we have X (k,t) = X (−k,t) for the field X (~r,t) and the other dynamical
ω ω ω
fields in this Lagrangian. Similarly the real valued coupling tensors f and g
can be expressed in reciprocal space as
1
f (ω,~r,r~′) = d3k d3k′f (ω,~k,k~′)eı~k·~r−ık~′·r~′
ij
(2π)3 ij
Z Z
1
g (ω,~r,r~′) = d3k d3k′g (ω,~k,k~′)eı~k·~r−ık~′·r~′ (8)
ij
(2π)3 ij
Z Z
which obey the following conditions
f (ω,~k,k~′) = f∗(ω,−~k,−k~′)
ij ij
g (ω,~k,k~′) = g∗ (ω,−~k,−k~′) (9)
ij ij
The number of independent variables can be recovered by restricting the
integrations to the half space k ≥ 0. The total Lagrangian (1)-(6) is then
z
obtained as
L(t) = L (t)+L (t)+L (t) (10)
res em int
∞ ′ ∞ ′
L (t) = dω d3k |X~˙ |2 −ω2|X~ |2 + dω d3k |Y~˙ |2 −ω2|Y~ |2
res ω ω ω ω
Z0 Z (cid:16) (cid:17) Z0 Z (cid:16) (11)(cid:17)
L (t) = ′d3k ε |A~˙|2 +ε |~k ϕ|2 − |~k ×A~|2 +ε ′d3k −ı~k ·A~˙ ϕ∗ +h.c
em 0 0 µ 0
0 !
Z Z (cid:16) (cid:17)
(12)
L (t) =
int
∞ ′ ′
− dω d3q d3p A~˙(~q,t)+ı~q ϕ(~q,t) ·f(ω,−~q,p~)·X~ (p~,t)+h.c
ω
Z0 Z Z h(cid:16) (cid:17) i
∞ ′ ′
− dω d3q d3p A~˙∗(~q,t)−ı~q ϕ∗(~q,t) ·f(ω,~q,p~)·X~ (p~,t)+h.c
ω
Z0 Z Z h(cid:16) (cid:17) i
∞ ′ ′
+ dω d3q d3p ı~q ×A~(~q,t) ·g(ω,−~q,~p)·Y~ (p~,t)+h.c
ω
Z0 Z Z h(cid:16) (cid:17) i
∞ ′ ′
+ dω d3q d3p −ı~q ×A~∗(~q,t) ·g(ω,~q,~p)·Y~ (p~,t)+h.c (13)
ω
Z0 Z Z h(cid:16) (cid:17) i
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′
where d3k implies the integration over the half space k ≥ 0 ( hereafter ,
z
Z ′
we apply the symbol d3k for the integration on the half space k ≥ 0 and
z
Z
d3k for the integration on the total reciprocal space). In the reciprocal
Z
space the total Lagrangian (10)- (13) do not involve the space derivatives of
the dynamical variables of the system and the classical equations of the mo-
tion of the system can be obtained using the principle of the Hamilton’s least
action, δ dt L(t) = 0. These equations are the Euler-Lagrange equations.
Z
For the vector potential A~(~k,t) and the scalar potential ϕ(~k,t) we find
d δL δL
− = 0 i = 1,2,3
dt δ A˙∗(~k,t)  δ A∗(~k,t)
i i
=⇒ µ εA~¨((cid:16)~k,t)+µ(cid:17)εı~k ϕ˙(~(cid:16)k,t)−~k ×(cid:17) ~k ×A~(~k,t) =
0 0 0 0
µ P~˙(~k,t)+ıµ ~k ×M~ (~k,t) (cid:16) (cid:17) (14)
0 0
d δL δL
− = 0
dt δ ϕ˙∗(~k,t)  δ ϕ∗(~k,t)
=⇒ −εı~k(cid:16)·A~˙(~k,t)(cid:17)+ε |~k|2ϕ(cid:16)(~k,t) =(cid:17)−ı~k ·P~(~k,t) (15)
0 0
~
for any wave vector k in the half space k ≥ 0 where
z
∞
P~(~k,t) = dω d3p f(ω,~k,~p)· X (p~,t) (16)
ω
Z0 Z
∞
M~ (~k,t) = dω d3p g(ω,~k,~p)· Y (p~,t) (17)
ω
Z0 Z
are respectively the spatial Fourier transforms of the electric and magnetic
polarization densities of the medium and it has been used from the relations
(9). Therefore in this method the polarization fields of the medium are
naturally concluded in terms of the coupling tensors f,g and the dynamical
variables of the E and M fields. Similarly the Euler-Lagrange equations for
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the fields X~ and Y~ for any vector ~k in the half space k ≥ 0 are easily
ω ω z
obtained as
d δL δL
− = 0 i = 1,2,3
dt δ X˙∗ (~k,t)  δ X∗ (~k,t)
ωi ωi
=⇒ X~¨ (~k,t(cid:16))+ω2 X~ (cid:17)(~k,t) =(cid:16)− d3q f(cid:17)†(ω,~q,~k)· A~˙(~q,t)+ı~qϕ(~q,t)
ω ω
Z (cid:16) (cid:17)
(18)
d δL δL
− = 0 i = 1,2,3
dt δ Y˙ ∗ (~k,t)  δ Y∗ (~k,t)
ωi ωi
=⇒ Y~¨ (~k,t(cid:16))+ω2 Y~ (cid:17)(~k,t) =(cid:16) d3q g†(ω(cid:17),~q,~k)· ı ~q ×A~(~q,t)
ω ω
Z (cid:16) (cid:17)
(19)
where f†andg† are the hermitian conjugate of the tensors f and g, respec-
tively.
3 Canonical quantization
Followingthestandardapproach, wechoosetheCoulombgauge~k·A~(~k,t) = 0
to quantize electromagnetic field. In this gauge the vector potential A~ is a
purely transverse field and can be decomposed along the unit polarization
vectors ~e λ = 1,2 which are orthogonal to each other and to the wave
λ~k
~
vector k.
2
~ ~ ~
A(k,t) = A (k,t)~e (20)
λ λ~k
λ=1
X
Although the vector potential is purely transverse, but the dynamical fields
X~ and Y~ may have both transverse and longitudinal parts and can be
ω ω
expanded along the three mutually orthogonal unit vectors ~e λ = 1,2
λ~k
8
and ~e = kˆ = ~k as
3~k |~k|
3
X~ (~k,t) = X (~k,t)~e
ω ωλ λ~k
λ=1
X
3
Y~ (~k,t) = Y (~k,t)~e (21)
ω ωλ λ~k
λ=1
X
Furthermore in Coulomb gauge the Euler- Lagrange equation (15) can be
~
used to eliminate the extra degree of freedom ϕ(k,t) from the Larangian of
the system
ı~k ·P~(~k,t)
~
ϕ(k,t) = − (22)
~
ε |k|2
0
The total Lagrangian(10)-(13) can now be rewritten in terms of the indepen-
dent dynamical variables A λ = 1,2 and X~ , Y~ λ = 1,2,3 which
λ ωλ ωλ
constitute completely the coordinates of the total system
∞ ′ 3
˙ ˙
L(t) = dω d3k |X~ |2 −ω2|X~ |2 +|Y~ |2 −ω2|Y~ |2
ωλ ωλ ωλ ωλ
Z0 Z Xλ=1(cid:16) (cid:17)
′ 2 |~k A |2 1 ′ |ı~k ·P~|2
+ d3k ε |A˙ |2 − λ − d3k
0 λ µ0 ! ε0 |~k|2
Z λ=1 Z
X
′ 2 2
+ d3k − A˙ ~e ·P~∗ + ı~k × A ~e ·M~ ∗ +h.c
λ λ~k λ λ~k
" ! ! #
Z λ=1 λ=1
X X
(23)
wherethepolarizationsP~ andM~ havebeendefined previouslyintermsofthe
dynamical variables of the E and M fields in equations (16) and (17). The
Lagrangian (23) can now be used to define the canonical conjugate momenta
~
of the system. For any wave vector k in half space k ≥ 0 this momenta are
z
9
defined as
δL
−D (~k,t) = = ε A˙ (~k,t)−~e ·P~(~k,t)
λ δ A˙∗(~k,t) 0 λ λ~k
λ
(cid:16) δL (cid:17)
Q (~k,t) = = X˙ (~k,t)
ωλ δ X˙∗ (~k,t) ωλ
ωλ
(cid:16) δL (cid:17)
Π (~k,t) = = Y˙ (~k,t) (24)
ωλ δ Y˙ ∗ (~k,t) ωλ
ωλ
(cid:16) (cid:17)
Thetotalsystemarequantizedcanonicallyinastandardmethodbyimposing
equal-time commutation relations between the coordinates of the system and
their conjugates variables as follows
A∗λ(~k,t) , −Dλ′(k~′,t) = ı~δλλ′δ(~k −k~′)
Xh ∗ωλ(~k,t) , Qω′λ′(k~′,t)i = ı~δλλ′δ(ω −ω′)δ(~k −k~′)
hY∗ωλ(~k,t) , Πω′λ′(k~′,t)i = ı~δλλ′δ(ω −ω′)δ(~k −k~′)
h i (25)
Using the Lagrangian (23) and the conjugates momenta introduced in (24)
the Hamiltonian of the total system can be written in the form
′ 2 |D −~e ·P~|2 |~kA |2 ′ |ı~k ·P~|2
H(t) = d3k λ λ~k + λ + d3k
Z λ=1 ε0 µ0 ! Z ε0|~k|2
X
′ 2
− d3k ı~k × A ~e ·M~ ∗ +h.c
λ λ~k
" ! #
Z λ=1
X
∞ ′ 3
+ dω d3k | Q |2 +ω2| X |2 +| Π |2 +ω2| Y |2
ωλ ωλ ωλ ωλ
Z0 Z Xλ=1(cid:16) (cid:17)
(26)
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