Using a phenomenological approach to field quantization, an expression for
the Keldysh function of photons between two planar interfaces (Casimir
geometry) is found for any stationary quantum state of the two bodies. The case
of one interface sliding against the other is considered in detail

Henkel, C.

Mkrtchian, V. E.

English

arXiv

Using a phenomenological approach to field quantization, an expression for the Keldysh function of photons between two planar interfaces (Casimir geometry) is found for any stationary quantum state of the two bodies. The case of one interface sliding against the other is considered in detail

V E Mkrtchian¹

C Henkel²

CiteSeerX

Armenian Journal of Physics, 2014, vol. 7, issue 3, pp. 106-112 
ON THE SPECTRAL DISTRIBUTION OF PHOTONS BETWEEN PLANAR 
INTERFACES 
V.E. Mkrtchian¹* and C. Henkel² 
¹Institute for Physical Research, Armenian Academy of Sciences, Ashtarak 0203, Republic of Armenia 
²Institute of Physics and Astronomy, University of Potsdam, Karl-Liebknecht-Str. 24/25, 14476 Potsdam, 
Germany 
*e-mail:vem@daad-alumni.de 
Received 8 May 2014 
 Using a phenomenological approach to field quantization, an expression for the Keldysh function of 
photons between two planar interfaces (Casimir geometry) is found for any stationary quantum state of 
the two bodies. The case of one interface sliding against the other is considered in detail.  
Key words: Casimir effect, planar interplace nonequilibrium, photon distribution 
 
1. Introduction 
 The simplest geometry in the electrodynamics of nanosystems is presumably the Casimir 
system: two half-spaces of different materials separated by a vacuum gap with two parallel 
planar interfaces. The gap has a width in the nano-range and provides the stage for propagating 
and evanescent waves emitted by the materials, thus shaping a rich plethora of Casimir physics: 
Casimir pressure, quantum friction, near-field radiative heat transfer, etc. All of these phenomena 
share a purely electrodynamic origin, and the basic theoretical quantity are the different 
components of Maxwell’s stress tensor. This tensor mirrors the quantum states of the 
electromagnetic field in the gap, and the latter is determined by macroscopic properties of the 
boundaries like their temperature distribution, current density and the relative motion of the 
interfaces. 
 In the case of stationary macroscopic conditions, a general formula for the distribution 
function of photons (Keldysh function) in the gap was recently found in Ref.[1], based on tools 
from nonequilibrium quantum field theory. This function provides a rather general description of 
the system and contains quantities (called fluctuation sources) which are determined by the 
above-mentioned macroscopic conditions of the system boundaries. In Ref.[1], the fluctuation 
sources were expressed by Keldysh functions evaluated on the interfaces which are quite 
difficult to handle. In this paper we find an alternative form for the sources that links them to the 
number of excitations of the compound system electromagnetic field+bulk medium, using the 
phenomenological theory of quantization of the electromagnetic field in a dissipative medium 
[2]. We also give explicit expressions for the Keldysh functions of a single interface system for 
further applications.  
Mkrtchian et al. || Armenian Journal of Physics, 2014, vol. 7, issue 3 
107 
 
 To avoid overloading of the paper with formulas, our notation and definitions follow those 
of Ref.[1]. 
2. Preliminaries 
 We consider two half-spaces of different media with parallel and homogeneous boundaries 
(what we know about boundaries are theirs reflection coefficients) located at z  νa/2  ν    . 
The boundaries are in stationary conditions: their temperatures are constant in time, and their 
relative motion is uniform and parallel to each other. We can then assume that the 
electromagnetic field (EMF) in the cavity ‐a/2    z    a/2  is stationary in time and 
statistically homogenous in the xy-plane. As a consequence, all relevant fields and correlation 
functions can be expanded in Fourier integrals with respect to frequency ω and wave vectors q   
qx,qy  along the interfaces. We use in the following the shorthand Ω    ω,q . In the 
Dzyaloshinskii gauge where φ   0, the transversality condition for the electric field can be used 
to eliminate the normal components of the vector potential in favor of the tangential ones (see [1] 
for details) which are conveniently expressed in the basis of s- and p-polarized fields of the lower 
interface (index λ   s,p). The EMF in the gap is not a closed system because of dissipativity of 
the enclosing boundaries, and the straightforward procedure of field quantization in vacuum 
cannot be applied. So, for a quantum description of the EMF in the gap, we have to resort to the 
theories of quantization in dissipative media. There are many such theories [2-5], but we shall 
take the simplest version called phenomenological quantization [2] which recently found its 
justification in the frame of the canonical quantization rules of quantum field theory in the 
dissipative medium [6]. We begin with the splitting of the operator of vector potential  into 
positive and negative parts in Fourier space: 
r,t , . .                                   (1) 
and then we postulate for the positive-frequency parts of the vector potential a representation in 
terms of current sources 
A Ω, z ∑ DR Ω; z, ν I Ω, ν                                        (2) 
at any point z of the gap. The surface currents I Ω, ν  here ‘live’on the two interfaces ν    .  
 In Eq.(2),  is the retarded Green function (RGF) of the EMF defined as  ђ   1) 
, ; , t t t , t , , t .                                   (3) 
The latter is a solution of an inhomogeneous wave equation with boundary conditions. The 
intensity of the radiation is distributed according to the Keldysh-Green function (KGF) which is 
a quantum expectation value of a symmetrized field correlation 
, t; , t , t , , t .                                              (4) 
ON THE SPECTRAL DISTRIBUTION || Armenian Journal of Physics, 2014, vol. 7, issue 3 
108 
 
 To find RGF and KGF in Eqs.(3, 4), we need averages of the commutator and 
anticommutator of A [Eq.(2)] with its Hermitian conjugate A , and then we come to compute 
averages of the commutator and anticommutator of the surface currents I ,I  introduced in 
Eq.(2). Our key assumption at this point is that we claim locality of the latter, writing 
I Ω, ν , I Ω , ν Ω Ω δ c Ω, ν ,                                 (5.a) 
I Ω, ν , I Ω , ν Ω Ω δ a Ω, ν ,                                (5.b) 
where we use the short-hand notation δ(Ω) ≡ (2π)3δ(q)δ(ω). Using Eqs.(5.a, 2) and the definition 
(3) of RGF, we come to an expression for the RGF that has the same structure as Eq.(D9) of 
Ref.[1]. Comparison of these two expressions gives the relation 
ω c Ω, ν ω cT Ω, ν iΓ Ω                                          (6) 
between the unknown matrix c in the r.h.s. of Eq.(5.a) and the matrix Γ of Eq.(3.26) in Ref.[1]. 
Taking into account the properties of Γ Ω  (a symmetric matrix and an odd function of Ω), we 
find for c  in Eq.(6) 
c Ω, ν iΓ Ω                                                                     (7) 
 With respect to the KGF (4), using Eqs.(1, 2), we can express it in the form of Eq.(4.21) of 
Ref.[1], where the photon sources P Ω, ν  are expressed by surface current anticommutators 
(5.b) according to 
iP Ω, ν ω a Ω, ν ω aT Ω, ν ,                                      (8) 
where −Ω = (-ω,q). Using a quantization procedure, we will show in the coming two sections 
that the photon sources P Ω, ν  are proportional to Γ Ω  matrices as well: 
P Ω, ν Ω Γ Ω ,                                                            (9) 
and the coefficients  (ν = ±) of proportionality are defined by excitation (occupation) numbers 
of the compound system electromagnetic field+body below the interface at νa/2. 
3. Quantization on the interface in rest 
 There are two interfaces in the problem: the lower one is in at rest ν     and the other one 
ν    is in a state of parallel uniform motion in the x-direction, say. We begin our 
consideration with the lower interface which is taken as a reference frame of the problem.  
 We introduce the annihilation operator f ,  of a Bose field which is associated with the 
elementary excitations of the composed system EMF+lower rmedium in the following way: 
Mkrtchian et al. || Armenian Journal of Physics, 2014, vol. 7, issue 3 
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I Ω, α Ω, f , Ω .                                                  (10) 
In Eq.(10), α is an as yet unknown matrix, and we suppose the canonical commutation rules for 
f ,  
f , Ω , f , Ω  δ δ(Ω-Ω')                                              (11.a) 
f , Ω , f , Ω  =  f , Ω , f , Ω  = 0                                    (11.b) 
We restrict our consideration considering a quantum state for the lower surface which is diagonal 
in the chosen basis and characterized by the excitation number N , (Ω): 
f , Ω f , Ω δ , δ(Ω-Ω')N , Ω                                         (12) 
Then, for the nonzero averages of commutator and anticommutator (5) of the surface currents on 
the lower interface, we have correspondingly 
c Ω, α Ω, α Ω, ,                                                (13.a) 
a Ω, α Ω, I 2N α Ω, ,                                     (13.b) 
where I is a 2 × 2 unit matrix and we introduced the diagonal matrix 
N
N , 0
0 N ,
 .                                                          (14) 
Taking into account that Γ  in Eq.(7) is diagonal, we also suggest the diagonality of α Ω,  in 
(13.a). This provides the anticommutator (13.b) and using Eq.(12), we find 
a Ω, i I 2N Ω Γ Ω .                                            (15) 
Insertion of Eqs.(15, 13.b) into Eq.(8) gives us expression (9) for ν     where  is defined in 
the same way as the photon number in free space [see Eq.(C4) of Ref.[1]]: 
Ω signωI 2 ω N Ω ω N Ω .                        (16) 
4. Quantization on the sliding interface 
 Using the transformation law for the surface currents and the properties of Fourier 
transforms under the Lorentz transformation of the space-time coordinates, we find the positive 
frequency part of the surface current operator in the laboratory frame I Ω, . It is expressed via 
corresponding operators in the reference frame of the moving interface [see Eq.(D2) of Ref.[1]] 
I Ω, O , Ω ω I Ω , ω   I Ω , ;   ω 0 ,           (17) 
ON THE SPECTRAL DISTRIBUTION || Armenian Journal of Physics, 2014, vol. 7, issue 3 
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where Ω ω ,  q , q  is related to the reference frame K  co-moving with the upper interface; 
it is connected with Ω = (ω,  ,    ) via the Lorentz transformation. The transformation matrix 
O Ω  in Eq.(17) is given in Eq.(D4) of Ref.[1].  
 Introducing a Bose field (with annihilation operator f ,    of excitations in the rest frame of 
the upper body analogous to Eqs.(10, 11) 
I Ω ; α Ω , f , Ω ,                                       (18.a) 
we find for the surface current anticommutator in K  
I Ω , , I Ω , iδ Ω Ω I 2N Ω Γ Ω .                 (18.b) 
And finally, using expression (17) for the positive-frequency part of the current operator in K, 
we find the anticommutator in the laboratory frame for ω ≥ 0 
a Ω, iO Ω Γ Ω OT,                                           (19.a) 
where 
Ω signω I 2 ω N Ω ω N Ω .                          (19.b) 
Using Eq.(8) for ν     we find for the photon sources (Ω,+) on the moving interface the 
expression 
P Ω, O Ω Γ Ω OT.                                            (20) 
Taking into account the transformation law of Γ  [Eq.(D11) of Ref.[1]], we come to the 
expression (9) for ν = + where the excitation number in the laboratory frame K is given by 
Ω O Ω O .                                               (21) 
Because of the invariance of the trace of the matrix under the similarity transformation in 
Eq.(21), the number of excitations in K and K are the same, and the relative motion only changes 
their polarization.  
 Obviously Eq.(20) recovers the result (7.3) of Ref.[1] in the case of thermal equilibrium in 
K′ . 
5. Examples of Keldysh Green functions 
 In this paragraph we give expressions for the KGF of Casimir system and for the single 
interface system for further applications. 
Mkrtchian et al. || Armenian Journal of Physics, 2014, vol. 7, issue 3 
111 
 
 The expression for KGF of the Casimir system with planar parallel boundaries is already 
written in Ref.[1], Eqs.(6.7–6.11). This is cumbersome enough not to repeat it here. We only 
give explicitly the matrices γ defined in Eq.(6.11) of Ref.[1] which characterize the fluctuation 
sources in KGF of the Casimir system.  
 Insertion of Eq.(9) into the definition of γ  , we find after some algebra 
γ e I Δ                                                          (22.a) 
for the sources in the lower interface and 
γ e I I R Δ Δ I R Δ                                    (22.b) 
in the upper, moving one. Here, we have put (22.b) 
I R R        
2iImR         ,
                                    (23) 
where propagating (evanescent) waves are defined by the observer-independent inequality 
q ω q 0  q 0 .  
 We get the KGF for a single, moving body (located in z    0) by applying a limiting 
procedure where the two points z,z      0 remain fixed, while the lower interface recedes to 
infinity (limit called (C) in Ref.[1]). This yields 
DK Ω; z, z I R Δ Δ I R e  
+ ,                        (24.a) 
where (Ω) is now interpreted as the average number of ‘up-propagating’photons that are 
incident on the moving body. If the upper interface is at rest, all matrices in (24.a) are diagonal, 
and we arrive at 
DK Ω; z, z e  
+  .                      (24.b) 
Finally, letting both bodies recede to infinity, with z, z ′ kept finite [limit(A) of Ref.[1]], we find 
the KGF of the EMF in free space of the EMF in free space 
DK Ω; z, z q e e Δ .                     (24.c) 
In the expressions (24),  are defining the number of free photons moving in opposite 
directions, which can be found in free space also by elementary plane wave quantization 
[Eq.(C4) of Ref.[1]]. 
ON THE SPECTRAL DISTRIBUTION || Armenian Journal of Physics, 2014, vol. 7, issue 3 
112 
 
6. Summary 
 We complete our considerations [1] of the photonic Keldysh function in the Casimir 
geometry of two plates by deriving explicit expressions for the fluctuation sources using 
phenomenological quantization of the electromagnetic field in a dissipative medium. We find, in 
particular, simple expressions for the Keldysh functions for a single interface system in an 
arbitrary stationary non-equilibrium state. 
Acknowledgments 
 We acknowledge financial support by the European Science Foundation (ESF) within the 
activity “New Trends and Applicationsof the Casimir Effect” (exchange grant 2847), and by the 
Deutsche Forschungsgemeinschaft (grant He 2849/4-1). 
References 
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2.  S Y Buhmann (2012) Dispersion Forces I – MacroscopicQuantum Electrodynamics and Ground-
State Casimir, Casimir–Polder and van der Waals Forces; in Springer Tracts in Modern Physics 
247 (Heidelberg: Springer). 
3. G S Agarwal, Phys. Rev. A 11 (1975) 230. 
4.  S M Rytov, Yu A Kravtsov, V I Tatarskii (1989) Elements of Random Fields, vol 3 of Principles 
of Statistical Radiophysics (Berlin: Springer). 
5.  S M Barnett (1997), The quantum optics of dielectrics, in Quantum fluctuations, ed S Reynaud, 
E Giacobino, and J Zinn-Justin (Amsterdam: Elsevier), pp 137–79. 
6.  T G Philbin, New J. Phys. 12 (2010) 123008 (Preprint quant-ph/1009.5005). 
 
 
 
 
 


On the spectral distribution of photons between planar interfaces

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