Canonical quantization of electromagnetic field in an anisotropic polarizable and magnetizable medium with spatial-temporal dispersion
aa r X i v : . [ qu a n t - ph ] J a n Canonical quantization of the electromagneticfield in an anisotropic polarizable andmagnetizable medium
M. Amooshahi ∗ Faculty of science, University of Isfahan ,Hezar Jarib Ave., Isfahan,Iran
October 24, 2018
Abstract
A fully canonical quantization of electromagnetic field is intro-duced in the presence of an anisotropic polarizable and magnetizablemedium . Two tensor fields which couple the electromagnetic fieldwith the medium and have an important role in this quantizationmethod are introduced. The electric and magnetic polarization fieldsof the medium naturally are concluded in terms of the coupling tensorsand the dynamical variables modeling the magnetodielectric medium.In Heisenberg picture, the constitutive equations of the medium to-gether with the Maxwell laws are obtained as the equations of motionof the total system and the susceptibility tensors of the medium arecalculated in terms of the coupling tensors. Following a perturbationmethod the Green function related to the total system is found andthe time dependence of electromagnetic field operators is derived.
Key words:
Canonical field quantization, Magnetodielectric medium,conductivity tensor, Susceptibility tensor, Coupling tensor, Constitu-tive equationPACS No: 12.20.Ds, 42.50.Nn ∗ [email protected] introduction One of the most important quantum dissipative systems is the quantizedelectromagnetic field in the presence of an absorbing polarizable medium. Inthis case there are mainly two quantization approaches, the phenomenologicalmethod [1]-[7] and the damped polarization model[8, 9]. The phenomenolog-ical scheme has been formulated on the basis of the fluctuation- dissipationtheorem [10]. In this method by adding a fluctuating noise term, that is thenoise 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 thefrequency domain, gives the electromagnetic field operators in terms of thenoise polarization field and the classical Green tensor. A set of bosonic op-erators is associated with the noise polarization which their commutationrelations are given in agreement with the fluctuation- dissipation theorem.This quantization scheme has been quite successful in describing some elec-tromagnetic phenomena in the presence of a lossy dielectric medium [11]-[15].The phenomenological approach has been extended to a lossy magnetic oranisotropic medium [16]-[18]. This formalism has also been generalized to anarbitrary linearly responding medium based on a spatially nonlocal conduc-tivity tensor [19].The damped polarization model to quantize electromagnetic field in a disper-sive dielectric medium [8, 9] is a canonical quantization in which the electricpolarization field of the medium applies in the Lagrangian of the total systemas a part of the degrees of freedom of the medium. The other parts of thedegrees of freedom of the absorbing medium are related to the dynamicalvariables of a heat bath describing the absorptivity feature of the medium.In this method the dielectric function of the medium is found in terms ofthe coupling function of the heat bath and the polarization field, so that itsatisfies the Kramers-Kronig relations [20]. This quantization method hasbeen generalized to an inhomogeneous medium[21].In the present work we generalize our previous model[22, 23] to an anisotropicdispersive magnetodielectric medium using a canonical approach. In thisformalism the medium is modeled with two independent collections of vec-tor fields. These collections solely constitute the degrees of freedom of themedium and it is not needed the electric and magnetic polarization fields tobe included in the Lagrangian of the total system as a part of the degreesof freedom of the medium as in the Huttner-Barnett model [8]. In fact the2ynamical fields modeling the dispersive medium are able to describe bothpolarizability 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 ananisotropic polarizable and magnetizable medium with spatial-temporal dis-persion briefly is discussed . In Sec. 3, applying the Lagrangian introducedin Sec. 2 a fully canonical quantization of both electromagnetic field andthe dynamical variable modeling the responding medium is demonstrated.Then in Sec. 4, the constitutive equations of the medium are obtained as theconsequences of the Heisenberg equations of the total system and the electricand magnetic susceptibility tensors of the medium are calculated in terms ofthe parameters applied in the theory . In Sec.5, it is shown that the Greenfunction 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 distinctionbetween polarization and magnetization is not possible. This paper is closedwith a summary and some concluding remarks in Sec.7 .
In order to present a fully canonical quantization of electromagnetic fieldin the presence of an anisotropic polarizable and magnetizable medium, wemodel the medium by two independent reservoirs. Each reservoir contains acontinium 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 itis not needed the electric and magnetic polarization fields of the medium tobe appeared explicitly in the Lagrangian of the total system as a part of thedegrees of freedom of the medium , but the contribution of the medium in theLagrangian of the total system is related only to the Lagrangian of the E and M fields and these fields completely describe the degrees of freedom of themedium. The presence of the ” E field” in the total Lagrangian is sufficientfor a complete description of both polarizability and the absorption of the3edium due to its electrically dispersive property. Also the ” M field” solelyis 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 res + L em + L int (1)where L res is the part related to the degrees of freedom of the medium andis the sum of the Lagrangians of the E and M fields L res = L e + L m (2)where L e = Z ∞ dω Z d r (cid:20)
12 ˙ ~X ω · ˙ ~X ω − ω ~X ω · ~X ω (cid:21) (3)and L m = Z ∞ dω Z d r (cid:20)
12 ˙ ~Y ω · ˙ ~Y ω − ω ~Y ω · ~Y ω (cid:21) (4)Here the fields ~X ω and ~Y ω are the dynamical variables of the E and M fields,respectively .In (1) L em is the contribution of the electromagnetic field in the La-grangian of the total system L em = Z d r " ε ~E − ~B µ (5)and L int is the part describing the interaction of the electromagnetic fieldwith the medium L int = Z ∞ dω Z d r Z d r ′ f ij ( ω, ~r, ~r ′ ) E i ( ~r, t ) ~X jω ( ~r ′ , t ) + Z ∞ dω Z d r Z d r ′ g ij ( ω, ~r, ~r ′ ) B i ( ~r, t ) ~Y jω ( ~r ′ , t ) (6)The contributions L e and L m in the Lagrangian L res are equivalent to theconsequences of diagonalization processes of the matter fields in the Huttner-Barnet model [8]. That is, L e ( L m ) is equivalent to the diagonalization4f 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 electricpolarization (magnetic polarization ) and the interaction term between theheat-bath B ( B ′ ) and the electric polarization (magnetic polarization). Inthe present approach modeling the medium , in a phenomenological way,with two independent set of oscillators the lengthy diagonalization processeshave been eliminated in the start of this quantization scheme. Particularlythe diagonalization processes may be more tremendous for an anisotropicmedium.In equations (5) and (6) ~E = − ∂ ~A∂t − ~ ∇ ϕ and ~B = ∇ × ~A are electricand magnetic fields respectively, where ~A and ϕ are the vector and the scalarpotentials. The tensors f and g in (6) are called the coupling tensors of themedium with electromagnetic field and for an inhomogeneous medium aredependent on the both position vectors ~r and ~r ′ . The coupling tensors arethe key parameters of this theory. As was mentioned above, it is not neededthe electric and magnetic polarization fields of the medium explicitly to beappeared in the total Lagrangian (1)-(6) as a part of degrees of freedom ofthe medium . As we will see, the electric polarization( magnetic polarization) of the medium is obtained in terms of the coupling tensor f ( g ) and thedynamical variables ~X ω ( ~Y ω ). Also the electric susceptibility tensor ( mag-netic susceptibility tensor) of the medium naturally will expressed in termsof the coupling tensor f ( g ). The coupling tensors f and g are appeared ascommon factors in both the noise polarization fields and the susceptibilitytensors of the medium, so that for the free space the susceptibility tensorstogether with the noise polarizations become identically zero and this quanti-zation scheme is reduced to the usual quantization of electromagnetic field infree space. Furthermore when the medium tends to a non-absorbing one, thecoupling 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 suchas in (6), it is the easiest way to work in the reciprocal space and write allthe fields and the coupling tensors f , g in terms of their spatial Fourier5ransforms. For example the dynamical variable ~X ω can be written as ~X ω ( ~r, t ) = 1 p (2 π ) ) Z d k ~X ω ( ~k, t ) e ı~k · ~r (7)Since we are concerned with real valued fields in the total Lagrangian (1)-(6),we have ~X ∗ ω ( ~k, t ) = ~X ω ( − ~k, t ) for the field ~X ω ( ~r, t ) and the other dynamicalfields in this Lagrangian. Similarly the real valued coupling tensors f and g can be expressed in reciprocal space as f ij ( ω, ~r, ~r ′ ) = 1(2 π ) Z d k Z d k ′ f ij ( ω, ~k, ~k ′ ) e ı~k · ~r − ı~k ′ · ~r ′ g ij ( ω, ~r, ~r ′ ) = 1(2 π ) Z d k Z d k ′ g ij ( ω, ~k, ~k ′ ) e ı~k · ~r − ı~k ′ · ~r ′ (8)which obey the following conditions f ij ( ω, ~k, ~k ′ ) = f ∗ ij ( ω, − ~k, − ~k ′ ) g ij ( ω, ~k, ~k ′ ) = g ∗ ij ( ω, − ~k, − ~k ′ ) (9)The number of independent variables can be recovered by restricting theintegrations to the half space k z ≥
0. The total Lagrangian (1)-(6) is thenobtained as L ( t ) = L res ( t ) + L em ( t ) + L int ( t ) (10) L res ( t ) = Z ∞ dω Z ′ d k (cid:16) | ˙ ~X ω | − ω | ~X ω | (cid:17) + Z ∞ dω Z ′ d k (cid:16) | ˙ ~Y ω | − ω | ~Y ω | (cid:17) (11) L em ( t ) = Z ′ d k ε | ˙ ~A | + ε | ~k ϕ | − | ~k × ~A | µ ! + ε Z ′ d k (cid:16) − ı~k · ˙ ~A ϕ ∗ + h.c (cid:17) (12) L int ( t ) = − Z ∞ dω Z ′ d q Z ′ d p h(cid:16) ˙ ~A ( ~q, t ) + ı~q ϕ ( ~q, t ) (cid:17) · f ( ω, − ~q, ~p ) · ~X ω ( ~p, t ) + h.c i − Z ∞ dω Z ′ d q Z ′ d p h(cid:16) ˙ ~A ∗ ( ~q, t ) − ı~q ϕ ∗ ( ~q, t ) (cid:17) · f ( ω, ~q, ~p ) · ~X ω ( ~p, t ) + h.c i + Z ∞ dω Z ′ d q Z ′ d p h(cid:16) ı~q × ~A ( ~q, t ) (cid:17) · g ( ω, − ~q, ~p ) · ~Y ω ( ~p, t ) + h.c i + Z ∞ dω Z ′ d q Z ′ d p h(cid:16) − ı~q × ~A ∗ ( ~q, t ) (cid:17) · g ( ω, ~q, ~p ) · ~Y ω ( ~p, t ) + h.c i (13)6here Z ′ d k implies the integration over the half space k z ≥ Z ′ d k for the integration on the half space k z ≥ Z d k for the integration on the total reciprocal space). In the reciprocalspace the total Lagrangian (10)- (13) do not involve the space derivatives ofthe 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 leastaction, δ Z dt L ( t ) = 0. These equations are the Euler-Lagrange equations.For the vector potential ~A ( ~k, t ) and the scalar potential ϕ ( ~k, t ) we find ddt δLδ (cid:16) ˙ A ∗ i ( ~k, t ) (cid:17) − δLδ (cid:16) A ∗ i ( ~k, t ) (cid:17) = 0 i = 1 , , ⇒ µ ε ¨ ~A ( ~k, t ) + µ ε ı~k ˙ ϕ ( ~k, t ) − ~k × (cid:16) ~k × ~A ( ~k, t ) (cid:17) = µ ˙ ~P ( ~k, t ) + ıµ ~k × ~M ( ~k, t ) (14) ddt δLδ (cid:16) ˙ ϕ ∗ ( ~k, t ) (cid:17) − δLδ (cid:16) ϕ ∗ ( ~k, t ) (cid:17) = 0= ⇒ − ε ı~k · ˙ ~A ( ~k, t ) + ε | ~k | ϕ ( ~k, t ) = − ı~k · ~P ( ~k, t ) (15)for any wave vector ~k in the half space k z ≥ ~P ( ~k, t ) = Z ∞ dω Z d p f ( ω, ~k, ~p ) · X ω ( ~p, t ) (16) ~M ( ~k, t ) = Z ∞ dω Z d p g ( ω, ~k, ~p ) · Y ω ( ~p, t ) (17)are respectively the spatial Fourier transforms of the electric and magneticpolarization densities of the medium and it has been used from the relations(9). Therefore in this method the polarization fields of the medium arenaturally concluded in terms of the coupling tensors f, g and the dynamicalvariables of the E and M fields. Similarly the Euler-Lagrange equations for7he fields ~X ω and ~Y ω for any vector ~k in the half space k z ≥ ddt δLδ (cid:16) ˙ X ∗ ωi ( ~k, t ) (cid:17) − δLδ (cid:16) X ∗ ωi ( ~k, t ) (cid:17) = 0 i = 1 , , ⇒ ¨ ~X ω ( ~k, t ) + ω ~X ω ( ~k, t ) = − Z d q f † ( ω, ~q, ~k ) · (cid:16) ˙ ~A ( ~q, t ) + ı~qϕ ( ~q, t ) (cid:17) (18) ddt δLδ (cid:16) ˙ Y ∗ ωi ( ~k, t ) (cid:17) − δLδ (cid:16) Y ∗ ωi ( ~k, t ) (cid:17) = 0 i = 1 , , ⇒ ¨ ~Y ω ( ~k, t ) + ω ~Y ω ( ~k, t ) = Z d q g † ( ω, ~q, ~k ) · (cid:16) ı ~q × ~A ( ~q, t ) (cid:17) (19)where f † andg † are the hermitian conjugate of the tensors f and g , respec-tively. Following the standard approach, we choose the Coulomb gauge ~k · ~A ( ~k, t ) = 0to quantize electromagnetic field. In this gauge the vector potential ~A is apurely transverse field and can be decomposed along the unit polarizationvectors ~e λ~k λ = 1 , ~k . ~A ( ~k, t ) = X λ =1 A λ ( ~k, t ) ~e λ~k (20)Although the vector potential is purely transverse, but the dynamical fields ~X ω and ~Y ω may have both transverse and longitudinal parts and can beexpanded along the three mutually orthogonal unit vectors ~e λ~k λ = 1 , ~e ~k = ˆ k = ~k | ~k | as ~X ω ( ~k, t ) = X λ =1 X ωλ ( ~k, t ) ~e λ~k ~Y ω ( ~k, t ) = X λ =1 Y ωλ ( ~k, t ) ~e λ~k (21)Furthermore in Coulomb gauge the Euler- Lagrange equation (15) can beused to eliminate the extra degree of freedom ϕ ( ~k, t ) from the Larangian ofthe system ϕ ( ~k, t ) = − ı~k · ~P ( ~k, t ) ε | ~k | (22)The total Lagrangian (10)-(13) can now be rewritten in terms of the indepen-dent dynamical variables A λ λ = 1 , ~X ωλ , ~Y ωλ λ = 1 , , L ( t ) = Z ∞ dω Z ′ d k X λ =1 (cid:16) | ˙ ~X ωλ | − ω | ~X ωλ | + | ˙ ~Y ωλ | − ω | ~Y ωλ | (cid:17) + Z ′ d k X λ =1 ε | ˙ A λ | − | ~k A λ | µ ! − ε Z ′ d k | ı~k · ~P | | ~k | + Z ′ d k " − X λ =1 ˙ A λ ~e λ~k ! · ~P ∗ + ı~k × X λ =1 A λ ~e λ~k ! · ~M ∗ + h.c (23)where the polarizations ~P and ~M have been defined previously in terms of thedynamical variables of the E and M fields in equations (16) and (17). TheLagrangian (23) can now be used to define the canonical conjugate momentaof the system. For any wave vector ~k in half space k z ≥ − D λ ( ~k, t ) = δLδ (cid:16) ˙ A ∗ λ ( ~k, t ) (cid:17) = ε ˙ A λ ( ~k, t ) − ~e λ~k · ~P ( ~k, t ) Q ωλ ( ~k, t ) = δLδ (cid:16) ˙ X ∗ ωλ ( ~k, t ) (cid:17) = ˙ X ωλ ( ~k, t )Π ωλ ( ~k, t ) = δLδ (cid:16) ˙ Y ∗ ωλ ( ~k, t ) (cid:17) = ˙ Y ωλ ( ~k, t ) (24)The total system are quantized canonically in a standard method by imposingequal-time commutation relations between the coordinates of the system andtheir conjugates variables as follows h A ∗ λ ( ~k, t ) , − D λ ′ ( ~k ′ , t ) i = ı ~ δ λλ ′ δ ( ~k − ~k ′ ) h X ∗ ωλ ( ~k, t ) , Q ω ′ λ ′ ( ~k ′ , t ) i = ı ~ δ λλ ′ δ ( ω − ω ′ ) δ ( ~k − ~k ′ ) h Y ∗ ωλ ( ~k, t ) , Π ω ′ λ ′ ( ~k ′ , t ) i = ı ~ δ λλ ′ δ ( ω − ω ′ ) δ ( ~k − ~k ′ ) (25)Using the Lagrangian (23) and the conjugates momenta introduced in (24)the Hamiltonian of the total system can be written in the form H ( t ) = Z ′ d k X λ =1 | D λ − ~e λ~k · ~P | ε + | ~kA λ | µ ! + Z ′ d k | ı~k · ~P | ε | ~k | − Z ′ d k " ı~k × X λ =1 A λ ~e λ~k ! · ~M ∗ + h.c + Z ∞ dω Z ′ d k X λ =1 (cid:16) | Q ωλ | + ω | X ωλ | + | Π ωλ | + ω | Y ωλ | (cid:17) (26)10 .1 Maxwell equations Using Heisenberg equations for the operators D λ and A λ , that is˙ A λ ( ~k, t ) = ı ~ h H , A λ ( ~k, t ) i = − D λ ( ~k, t ) − ~e λ~k · ~P ( ~k, t ) ε ˙ D λ ( ~k, t ) = ı ~ h H , D λ ( ~k, t ) i = | ~k | µ A λ ( ~k, t ) − ~e λ~k · (cid:16) ı~k × ~M ( ~k, t ) (cid:17) (27)Maxwell equations can be obtained as is expected. Multiplying both sides ofEqs (27) in the polarization unit vector ~e λ~k and then summation on polar-ization index λ = 1 , ~D ( ~k, t ) = ε ~E ( ~k, t ) + ~P ( ~k, t ) (28)˙ ~D ( ~k, t ) = ı~k × ~H (29)where D = X λ =1 ˙ D λ ~e λ~k plays the role of the displacement field, ~E = − ˙ ~A − ı~kϕ = − ˙ ~A − ˆ k (ˆ k · ~P ) ε is the total electric field and ~H ( ~k, t ) = ı~k × ~Aµ − ~M isthe magnetic induction field. It is well known that the constitutive equations of a responding medium arethe consequences of the interaction of the medium with electromagnetic field.Any quantization method such as this theory, in which the medium enter di-rectly in the process of quantization and the interaction of the medium withthe vacuum field is explicitly given, must be able to give the constitutiveequations of the medium using the Heisenberg equations of the total system.Also the susceptibility tensors of the medium which are a measure for thepolarizability of the medium should be specified in terms of the parametersdescribing the interaction of the medium with electromagnetic field. In thissection, we find the constitutive equations of the medium using the Heisen-berg equations of the dynamical fields modeling the medium. As we do this,11he electric and magnetic susceptibility tensors of the medium are naturallyfound in terms of the coupling tensors f and g . In the Heisenberg picture,using the commutation relations (25) and the total Hamiltonian (26), theequations of motion for the canonical variables X ωλ , Q ωλ λ = 1 , , k z ≥ X ωλ ( ~k, t ) = ı ~ [ H, X ωλ ( ~k, t )] = Q ωλ ( ~k, t )˙ Q ωλ ( ~k, t ) = [ H, Q ωλ ( ~k, t )] = − ω X ωλ ( ~k, t ) + Z d qf ∗ ij ( ω, ~q, ~k ) E i ( ~q, t ) e jλ~k (30)where ~E ( ~q, t ) = − ˙ ~A ( ~q, t ) − ˆ q (cid:16) ˆ q · ~P ( ~q, t ) (cid:17) ε is the Fourier transform of thetotal electric field. Multiplying both sides of Eqs. (30) in the mutuallyorthogonal unit vectors ~e λ~k and then summation on λ = 1 , , X λ =1 e iλ~k e jλ~k = δ ij we find¨ ~X ( ~k, t ) + ω ~X ( ~k, t ) = Z d q f † ( ω, ~q, ~k ) · ~E ( ~q, t ) (31)This equation can be integrated formally as ~X ω ( ~k, t ) = ~Q ω ( ~k,
0) sin ωtω + ~X ω ( ~k,
0) cos ωt + Z t dt ′ sin ω ( t − t ′ ) ω Z d q f † ( ω, ~q, ~k ) · ~E ( ~q, t ′ ) (32)Now, by substituting ~X ω ( ~k, t ) from (32) in the definition of electric polariza-tion density given by (16), we find the constitutive equation of the mediumin reciprocal space relating the electric polarization to the electric field ~P ( ~k, t ) = ~P N ( ~k, t ) + ε Z | t | dt ′ Z d q χ e ( ~k, ~q, | t | − t ′ ) · ~E ( ~q, ± t ′ ) (33)where the upper (lower) sign corresponds to t > t < χ e isthe spatial Fourier transform of the electric susceptibility tensor χ e ( ~k, ~q, t ) = ε R ∞ dω sin ωtω R d p f ( ω, ~k, ~p ) · f † ( ω, ~q, ~p ) t > t ≤ f in terms ofthe temporal Fourier transform of χ e using a type of eigenvalue problem[19].It must be pointed out that for a given χ e the coupling tensor f satisfying therelation (34) is not unique. In fact if f satisfy (34), for a given susceptibilitytensor ,then f ′ ( ω, ~k, ~q ) = Z d p ′ f ( ω, ~k, ~p ′ ) · A ( ω, ~q, ~p ′ ) (35)also satisfy the (34) where A is a tensor with orthogonality condition Z d p A ( ω, ~p, ~p ′ ) · A † ( ω, ~p, ~p ′′ ) = Iδ ( ~p ′ − ~p ′′ ) (36)Although for a given χ e various choices of the tensor f satisfying ( 34) af-fect the space-time dependence of electromagnetic field operators, but all ofthese choices are equivalent and the commutation relations between electro-magnetic field operators remain unchanged [24]. That is, the commutationrelations between electromagnetic operators and the physical observables fi-nally are dependent only on the given susceptibility tensor and not on thecoupling functions f , g .In (33), ~P N is the spatial Fourier transform of the noise electric polarizationdensity which is expressed in terms of the the dynamical variables of the ” E field” at t = 0 ~P N ( ~k, t ) = Z ∞ dω Z d p f ( ω, ~k, ~p ) · (cid:18) ~Q ω ( ~p,
0) sin ωtω + ~X ω ( ~p,
0) cos ωt (cid:19) (37)In a similar fashion the constitutive equation relating the magnetic polariza-tion density of the medium to the magnetic field B ( ~q, t ) = ı~q × A ( ~q, t ) can beobtained straightforwardly using the Heisenberg equations for the conjugatedynamical variables Y ωλ and Π ωλ as the form ~M ( ~k, t ) = ~M N ( ~k, t ) + 1 µ Z | t | dt ′ Z d q χ m ( ~k, ~q, | t | − t ′ ) · ~B ( ~q, ± t ′ ) (38)where the χ m is the spatial Fourier transform of the magnetic susceptibilitytensor which is written in terms the coupling tensor g as χ m ( ~k, ~q, t ) = µ R ∞ dω sin ωtω R d p g ( ω, ~k, ~p ) · g † ( ω, ~q, ~p ) t > t ≤ ~M N ( ~k, t ) = Z ∞ dω Z d p g ( ω, ~k, ~p ) · (cid:18) ~ Π ω ( ~p,
0) sin ωtω + ~Y ω ( ~p,
0) cos ωt (cid:19) (40)is the noise magnetic polarization density. Therefore modeling an anisotropicmagnetodielectric medium with two independent collections of harmonic os-cillators, that is, the E and M field, we successfully have constructed afully canonical quantization of electromagnetic field in the presence of such amedium, so that the Maxwell equations together with the constitutive equa-tions of the medium have been obtained from the Heisenberg equations ofthe total system. In this method the susceptibility tenors of the medium areconcluded in terms of the coupling tensors which are the key parameters ofthis theory and are describing the coupling of the electromagnetic field withthe medium.The Equations (34) and (39) relate the parameters of this theory, that isthe coupling tensors f and g , to the physical quantities χ e and χ m . If thecoupling tensors f and g are specified, so that the right hands of (34) and(39) become identical to the electric and magnetic susceptibility tensors ofthe medium, then according to the constitutive equations (33) and (38) theoperators ~P and ~M defined by (16) and (17) are respectively the polarizationand magnetization of the medium.There are media for which the polarization ( the magnetization) is depen-dent on both the electric and magnetic field. This quantization scheme canbe generalized for such media if in the interaction Lagrangian (6) each ofthe dynamical variables ~X ω and ~Y ω is interacted with both the electric andmagnetic field. Our goal for this section is to determine the complete expressions of theelectromagnetic field operators for both negative and positive times. TheHeisenberg equations of the total system, that is the Maxwell equation (29)together with the constitutive equations (28), (33) and (38), constitute a14et of coupled linear equations which can be solved in terms of the initialconditions by the temporal Laplace transformation. This technique has beenused previously in the damped polarization model[21]. For an arbitrary time-dependent operator Γ( t ) the backward and forward Laplace transformations,denoted respectively, by Γ b ( s ) and Γ f ( ρ ) are defined asΓ b ( s ) = Z ∞ dt e − st Γ( − t )Γ f ( s ) = Z ∞ dt e − st Γ( t ) (41)Carrying out the backward and forward Laplace transformations of Eqs.(28), (29), (33) , (38) and ı~k × ~E ( ~k, t ) = − ˙ ~B ( ~k, t ) and then their combination,for the electric field operator we find Z d q h ~k × ˜ µ ( ~k, ~q, s ) (cid:16) ~q × ~E b ( ~q, s ) (cid:17)i − s c Z d q ˜ ε ( ~k, ~q, s ) ~E b ( ~q, s ) = ~J bN ( ~k, s ) Z d q h ~k × ˜ µ ( ~k, ~q, s ) (cid:16) ~q × ~E f ( ~q, s ) (cid:17)i − s c Z d q ˜ ε ( ~k, ~q, s ) ~E f ( ~q, s ) = ~J fN ( ~k, s )(42)where ˜ ε ( ~k, ~q, s ) = δ ( ~k − ~q ) I + ˜ χ e ( ~k, ~q, s )˜ µ ( ~k, ~q, s ) = δ ( ~k − ~q ) I − ˜ χ m ( ~k, ~q, s ) (43)are respectively the permitivity and permeability tensors of the medium inLaplace language. The source terms in the right hand of the inhomogeneousEqs.(42) are the backward and forward Laplace transformations of the noisecurrent density and are expressed in terms of initial conditions of the dynam-ical variables of the total system at t = 0 as ~J bN ( ~k, s ) = µ s ~P bN ( ~k, s ) − µ s ı~k × M bN ( ~k, s ) + ı~k × Z d q ˜ µ ( ~k, ~q, s ) ~B ( ~q, − µ s ~D ( ~k, ~J fN ( ~k, s ) = µ s ~P fN ( ~k, s ) + µ s ı~k × M fN ( ~k, s ) − ı~k × Z d q ˜ µ ( ~k, ~q, s ) ~B ( ~q, − µ s ~D ( ~k,
0) (44)15qs. (42) can be rewritten in a compact form as Z d q Λ( ~k, ~q, s ) · ~E b ( ~q, s ) = ~J bN ( ~k, s ) Z d q Λ( ~k, ~q, s ) · ~E f ( ~q, s ) = ~J fN ( ~k, s ) (45)with Λ ij ( ~k, ~q, s ) = ε iδγ ε αβj k δ q β ˜ µ γα ( ~k, ~q, s ) − s c ˜ ε ij ( ~k, ~q, s ) (46)From (45) we see that the backward and forward Laplace transformations ofthe electric field operator satisfy an inhomogeneous equation with a sourceterm. To solve such equations it is common to use the Green functionmethod. The suitable Green function associated to the Eq. (45) is definedas the solution of an algebraic equation as follows Z d q Λ( ~k, ~q, s ) · G ( ~q, ~p, s ) = I δ ( ~k − ~p ) (47)Accordingly the solution of Eqs. (45) in terms of the Green function definedin (47) can be written as ~E b ( ~k, s ) = Z d p G ( ~k, ~p, s ) · ~J bN ( ~p, s ) ~E f ( ~k, s ) = Z d p G ( ~k, ~p, s ) · ~J fN ( ~p, s ) (48)In the special case of a homogeneous medium for which the electric andmagnetic susceptibility tensors are translationally invariant, that is when thetensors χ e ( ~r, ~r ′ , t ) and χ m ( ~r, ~r ′ , t ) are a function of the difference ~r − ~r ′ andso is then, the permitivity and permeability tensors ε ( ~r, ~r ′ , t ) and µ ( ~r, ~r ′ , t ),we deduce ˜ ε ( ~k, ~q, s ) = (cid:16) I + ˜ χ e ( ~k, s ) (cid:17) δ ( ~k − ~q )˜ µ ( ~k, ~q, s ) = (cid:16) I − ˜ χ m ( ~k, s ) (cid:17) δ ( ~k − ~q ) (49)and therefore from the Eqs. (46) and (47) we find G ( ~k, ~p, s ) = T − ( ~k, s ) δ ( ~k − ~p ) T ij = (cid:20) ε iδγ ε αβj k δ k β (cid:16) δ γα − ˜ χ mγα ( ~k, s ) (cid:17) − s c (cid:16) δ ij + ˜ χ eij ( ~k, s ) (cid:17)(cid:21) (50)16here it should be summed on the indices α, β, γ, δ in the first term in thebracket. For a general medium it may be impossible to solve Eq.(47) exactlyto obtain the Green function G . However one can use a suitable iterationmethod and find the Green function up to an arbitrary accuracy which maybe useful for media with considerably weak polarizability. Let us write thetensor Λ( ~k, ~q, s ) as the sum of two partsΛ( ~k, ~q, s ) = Λ (0) ( ~k, ~q, s ) + Λ (1) ( ~k, ~q, s ) (51)whereΛ (0) ij ( ~k, ~q, s ) = (cid:18) k i k j − δ ij k − s c δ ij (cid:19) δ ( ~k − ~q ) ≡ L ( ~k, s ) δ ( ~k − ~q ) (52)is the tensor Λ in free space, that is when there is no medium andΛ (1) ij ( ~k, ~q, s ) = − ε iδγ ε αβj k δ q β ˜ χ mγα ( ~k, ~q, s ) − s c ˜ χ eij ( ~k, ~q, s ) (53)is the part due to the presence of the medium, that is the effect of themedium in the total tensor Λ. Then, following the Born approximationmethod previously used in the scattering theory [25], the Green function G can be expressed as a series as follows G ( ~k, ~p, s ) = G (0) ( ~k, ~p, s ) + G (1) ( ~k, ~p, s ) + G (2) ( ~k, ~p, s ) + · · · (54)where G (0) ( ~k, ~p, s ) = L − ( ~k, s ) δ ( ~k − ~p ) (55)is the Green function for the free space and G ( n ) ( ~k, ~p, s ) n = 1 , , · · · arefound from the following recurrence relation Z d q Λ (0) ( ~k, ~q, s ) · G ( n ) ( ~q, ~p, s ) = − Z d q Λ (1) ( ~k, ~q, s ) · G ( n − ( ~q, ~p, s )= ⇒ G ( n ) ( ~k, ~p, s ) = − L ( − ( ~k, s ) Z d q Λ (1) ( ~k, ~q, s ) · G ( n − ( ~q, ~p, s ) (56)where we have used Eq.(52). Using the series (54) and the recurrence relation(56) one can obtain the Green function G up to an arbitrary accuracy. Inn’th order approximation it can be neglected from the terms G ( j ) j ≥ n + 1in the series (54) which may be useful for media with considerably weak17olarizability.Now, having the Green function G , the time dependence of electric fieldoperator for negative and positive times can be obtained by inverse Laplacetransformations of ~E b and ~E f , respectively. Carrying out the inverse Laplacetransformation of Eq. (48) for t < ~E ( ~k, t ) = 12 π Z + ∞−∞ dω e − ıωt Z d p G ( ~k, ~p, ıω + 0 + ) · ~J bN ( ~p, ıω + 0 + ) (57)where 0 + is an arbitrarily small positive number. For t > ~E f is ~E ( ~k, t ) = 12 π Z + ∞−∞ dω e − ıωt Z d p G ( ~k, ~p, − ıω +0 + ) · ~J fN ( ~p, − ıω +0 + ) (58)Because the Laplace transformations of the susceptibility tensors are analyticfunctions respect to the variable s in any point of the half-plane Re [ s ] ≥ ω -dependence integrand in Eq. (57) also is analytic in thehalf-plane Im [ ω ] ≤
0. Accordingly, the integral over ω in (57) vanishes forthe positive times . Likewise the integral in (58) is zero for t <
0. Thereforewe can combine the two expressions (57) and (58) into a single one and writethe full time-dependence of electric field for all times as ~E ( ~k, t ) = 12 π Z + ∞−∞ dω e − ıωt Z d p h G ( ~k, ~p, ıω + 0 + ) · ~J bN ( ~p, ıω + 0 + )+ G ( ~k, ~p, − ıω + 0 + ) · ~J fN ( ~p, − ıω + 0 + ) i (59)It is remarkable that using the asymptotic behavior of the Green function G and the susceptibility tensors for large | ω | it can be verified straightforwardly,the integral in (59) reduces to ~E ( ~k,
0) at t = 0 and hence this equation giveus correctly the time dependence of the electric field for all times. Finallyhaving the electric field operator, we can obtain the magnetic field using ı~k × ~E ( ~k, t ) = ˙ ~B ( ~k, t ) and then the polarization densities employing theconstitutive Eqs.(33) and (38). 18 The magnetodielectric media with only oneresponse equation
In the previous sections we have assumed the magnetodielectric mediumunder consideration is one for which the distinction between polarizationand magnetization physically is possible and the total current induced inmedium can be separated into electric and magnetic parts. It is remarkablethat the use of polarization and magnetization , separately, to describe theresponse of a medium to electromagnetic field is useful for media which arenot spatially dispersive or which have a particular form of spatial dispersion[26]. There are in general media for which it is not clear how the separation ofthe total current induced in medium is to be made into electric and magneticparts. The disturbances, caused by the electromagnetic field, in such mediaare described completely in terms of only a response equation relating thetotal current induced in medium to the electric field [26]. The quantizationmethod outlined in the previous sections can cover such cases applying theLagrangian L ( t ) = Z ∞ dω Z d r (cid:20)
12 ˙ ~X ω · ˙ ~X ω − ω ~X ω · ~X ω (cid:21) + Z d r " ε ~E − ~B µ + Z ∞ dω Z d r Z d r ′ f ij ( ω, ~r, ~r ′ ) E i ( ~r, t ) ~X jω ( ~r ′ , t ) (60)which is the same as the Lagrangian (1)-(6) with the exceptional that, nowthe medium is modeled by a single set of three dimensional harmonic oscil-lators ~X ω . In this case the Euler- Lagrange equation (14) takes the form ddt δLδ (cid:16) ˙ A ∗ i ( ~k, t ) (cid:17) − δLδ (cid:16) A ∗ i ( ~k, t ) (cid:17) = 0 i = 1 , , ⇒ µ ε ¨ ~A ( ~k, t ) + µ ε ı~k ˙ ϕ ( ~k, t ) − ~k × (cid:16) ~k × ~A ( ~k, t ) (cid:17) = µ ˙ ~P ( ~k, t )(61)where now ~P ( ~k, t ) = Z ∞ dω Z d p f ( ω, ~k, ~p ) · X ω ( ~p, t ) (62)19an no longer be interpreted as the electric polarization, but is related tothe total current induced in the medium by j ( ~k, t ) = ˙ ~P ( ~k, t ). Following thecanonical quantization similar to the previous sections we obtain the onlyresponse equation of the medium as ~P ( ~k, t ) = ~P N ( ~k, t ) + ε Z | t | dt ′ Z d q χ ( ~k, ~q, | t | − t ′ ) · ~E ( ~q, ± t ′ ) (63)instead of the two constitutive equations (33) and (40), where ~P N ( ~k, t ) isgiven by (37) and the response tensor χ ( ~k, ~q, t ) = ε R ∞ dω sin ωtω R d p f ( ω, ~k, ~p ) · f † ( ω, ~q, ~p ) t > t ≤ ∂χ∂t = σ . This is oneof the two alternative descriptions of the response of a magnetodielectricmedium for which the distinction between polarization and magnetization isnot possible [26]. Finally following the same calculations as before sectionsone can obtain the space -time dependence of electromagnetic field operatorsin terms of the response tensor χ and the dynamical variables ~X ω at t = 0.The calculations in this case are precisely the same as before sections exceptthat the quantities ~Y ω , ~ Π ω , ~M , χ m , g are identically zero. By modeling an anisotropic polarizable and magnetizable medium with twocontinuous collections of three dimensional vector fields, we have presented afully canonical quantization for electromagnetic field in the presence of sucha medium. In this model the polarization fields of the medium did not enterdirectly in the Lagrangian of the total system as a part of the degrees offreedom of the medium, but the space-dependent oscillators modeling themedium solely was sufficient to describe both polarizability and the absorp-tion of the medium. The interaction of the medium with the electromagneticfield explicitly was given and both the Maxwell’s laws and the constitutiveequations of the medium were obtained as the consequences of the Heisen-berg equations of total system. The electric and magnetic polarization fields20f the medium could be deduced naturally in terms of the dynamical fieldsmodeling the medium. Some space dependent real valued tensor couplingthe medium with electromagnetic field were introduced. The coupling tensorhad an important role in this quantization scheme, so that the electric andmagnetic susceptibility tensors of the medium were determined in terms ofthe coupling tensor. Also the noise polarizations were expressed in termsof the coupling tensor and the dynamical variables describing the degreesof freedom of the medium at t = 0. In the free space and in the case of anon-absorbing medium the coupling tensor and the noise polarizations tendto zero and this quantization method is reduced to the usual quantization inthese limiting cases. As a tool, we have used the reciprocal space to quan-tize the total system. It was shown the temporal backward and forwardLaplace transforms of the electric field obey some algebraic equations withsource terms that could be determined in terms of the dynamical variablesof the system at t = 0. By introducing the Green function of these algebraicequations we were able to determine the Green function up to an arbitraryaccuracy using a perturbation method which may be useful in some real caseswith weak polarizability. The time-dependence of electric field was derivedfor both negative and positive times. Finally the model was modified tothe case of a medium for which the distinction between the polarization andmagnetization is not possible. References [1] T. Gruner, D. G. Welsch, Phys. Rev. A 51, 3246 (1995).[2] R. Matloob, R. Loudon, S. M. Barnett, J. Jeffers, Phys. Rev. A 52, 4823(1995).[3] T. Gruner, D. G. Welsch, Phys. Rev. A 53, 1818 (1996).[4] R. Matloob, R. Loudon, Phys. Rev. A 53, 4567 (1996).[5] T. Gruner, D. G. Welsch, Phys. Rev. A 54, 1661 (1996).[6] H. T. Dung, L. Kn¨oll, D. G. Welsch, Phys. Rev. A 57, 3931(1998)[7] S. Scheel, L. Kn¨oll, D. G. Welsch, Phys. Rev. A 58, 700(1998)[8] B. Huttner, S. M. Barnett, Phys. Rev. A 46, 4306 (1992).219] B. Huttner, S. M. Barnett, Europhys. Lett. 18, 487 (1992).[10] U. Weiss, Dissipative Quantum Systems (World Scientific 1993).[11] S. M. Barnett, B. Huttner, R. Loudon, R. Matloob, J. Phys. B 29,3763(1996)[12] S. Scheel, L. Kn¨oll, D. G. Welsch, S. M. Barnett, Phys. Rev. A 60,1590(1999).[13] S. Scheel, L. Kn¨oll, D. G. Welsch, Phys. Rev. A 60, 4094 (1999).[14] H. T. Dung, L. Kn¨oll, D. G. Welsch, Phys. Rev. A 62, 053804(2000)[15] H. T. Dung, L. Kn¨oll, D. G. Welsch, Phys. Rev. A 66, 063810(2002)[16] L. Kn¨oll, S. Scheel, D. G. Welsch, in Coherence and Statistics ofphothons and atoms , edited by J. Peˇrina ( Wiley, Newyork, 2001).[17] A. Lukˇs, V. Peˇrinovˇa, in progress in optics , Edited by E. Wolf ( North-Holand, Amsterdam, 2002 Vol. 43, p. 295).[18] H. T. Dung, S. Y. Buhmann, L. Kn¨oll, D. G. Welsch, Phys. Rev. A 68,043816 (2003).[19] C. Raabe, S. Scheel, D. G. Welsch, Phys. Rev. A 75, 053813 (2007).[20] L. D. Landau, E. M. Lifshitz,
Electrodynamics of Continuous Media (oxford: Pergamon, 1977).[21] L. G. Suttorp, M. Wubs, Phys. Rev. A 70, 013816 (2004)[22] F. Kheirandish, M. Amooshahi, Phys. Rev. A 74, 042102(2006).[23] M. Amooshahi, F. Kheirandish, Phys. Rev. A 76, 062103(2006).[24] M. Amooshahi, F. Kheirandish, Modd. Phys. Lett. A 23, No. 26 (2008)[25] J. J. Sakurai,
Modern quantum mechanics , (Adisson- wesley 1985).[26] D. B. Melrose, R. C. McPhedran,