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Keywords:

  • Casimir effect;
  • van der Waals interaction;
  • quantum friction;
  • nonequilibrium electrodynamics of nanosystems

Abstract

  1. Top of page
  2. Abstract
  3. 1 Introduction
  4. 2 Preliminaries
  5. 3 Retarded Green functions
  6. 4 Non-equilibrium action
  7. 5 The large distance limit
  8. 6 Keldysh–Green functions
  9. 7 Example: sliding interfaces
  10. 8 Conclusions
  11. Appendix A
  12. Appendix B
  13. Appendix C
  14. Appendix D
  15. Appendix E
  16. Acknowledgments
  17. References

The electromagnetic field in a typical geometry of the Casimir effect is described in the Schwinger–Keldysh formalism. The main result is the photon distribution function (Keldysh Green function) in any stationary state of the field. A two-plate geometry with a sliding interface in local equilibrium is studied in detail, and full agreement with the results of Rytov fluctuation electrodynamics is found.

1 Introduction

  1. Top of page
  2. Abstract
  3. 1 Introduction
  4. 2 Preliminaries
  5. 3 Retarded Green functions
  6. 4 Non-equilibrium action
  7. 5 The large distance limit
  8. 6 Keldysh–Green functions
  9. 7 Example: sliding interfaces
  10. 8 Conclusions
  11. Appendix A
  12. Appendix B
  13. Appendix C
  14. Appendix D
  15. Appendix E
  16. Acknowledgments
  17. References

In his seminal article 65 years ago, Casimir formulated a physical problem [1] which has had a tremendous influence on physics. His pioneering analysis of the physical consequences of field quantization under external, macroscopic boundary conditions is still of current relevance. Indeed, it turned out to be one of the most prolific ideas in modern theoretical physics. Having a pure quantum background and being closely related to classical physics, the Casimir effect is a universal and wide-spread phenomenon that can be found on all scales in the Universe.

One of the fascinating aspects of the Casimir effect is its simplicity, i.e., there are theoretical models that are amenable to exact solutions. Leaving aside many important features of modern Casimir physics [2], we consider the simple system of Ref. [1] with a slight generalization. Namely, our setting consists of two half-spaces of different materials with a plane-parallel gap in between. What we know about the two media are their reflection coefficients (a matrix in general) and macroscopic conditions like temperature distributions, macroscopic current densities, distance and relative motion parallel to the interfaces. We assume that these conditions are maintained stationary by external (generalized) forces. We want to compute correlation functions of the electromagnetic field in the gap. These yield for example the average energy density of the field and the pressure exerted on the two bodies as components of the stress (energy-momentum) tensor. We focus in particular on symmetrized correlations, also known as Keldysh-Green functions (KGF) in quantum kinetics, and are able to derive them under rather general circumstances in a stationary Casimir geometry. We thus establish a natural non-equilibrium extension of Casimir's results.

Our approach starts from a simple observation which is a common feature among all manifestations of the Casimir effect: the interaction between the two interfaces disappears when their distance becomes large enough. On quite general grounds, we may therefore conclude that the electromagnetic field which we observe in any Casimir system results from several fields that originate in the material of the different half-spaces and in the vacuum gap between them. The distance dependence of the Casimir interaction is such that these fields become independent (uncorrelated) as the interfaces are infinitely far away. This fact is the cornerstone of our analysis: we use the large-distance limit to express the KGF in a two-plate setting in terms of KGFs of independent half-spaces.

In this paper, we relax the assumption that the two plates share the same temperature and state of motion. This makes it impossible to describe the field between the plates as being in thermal equilibrium. The appropriate tool to calculate photon correlation functions is then the Schwinger-Keldysh technique[3, 4] of nonequilibrium processes. The application of the Keldysh formalism to the field of Casimir-Van der Waals interactions has a relatively short history. The first application, to our knowledge, is due to Janowicz, Reddig and Holthaus [5] in calculations of the electromagnetic heat transfer between two bodies at different temperature. Sherkunov used the non-equilibrium technique extensively for dispersion interactions involving excited atoms and excited media [6]. A general expression for the electromagnetic force on an atom, in terms of the KGFs of atom and radiation field, was found a few years ago by one of the present authors [7].

The physical processes behind these field-mediated interactions are the multiple reflections of photons between the interfaces and their tunneling from one body to the other, which follow from the boundary conditions for the electromagnetic field on the interfaces. To include these boundary conditions, we use an effective action in the Schwinger-Keldysh technique with auxiliary fields and evaluate generating functionals for KGFs by performing path integrals [8]. Path integral approaches for the Casimir effect were introduced by Bordag, Robaschik and Wieczorek [9]. Li and Kardar considered the interaction between bodies mediated by a fluid with long range correlations [10, 11]. They applied a path integral technique to include arbitrarily deformed bodies on which any kind of boundary condition can be implemented. This feature makes the approach amenable to a perturbative analysis of any deformed ideally conducting surfaces [12, 13]. Further on, Emig and Büscher [14] used the optical extinction theorem [15] to reformulate the boundary conditions for a vacuum-dielectric interface in integral form that depends only on the fields on the vacuum side (i.e., in the gap between two bodies). They then derived with path integral techniques an effective Gaussian action for the photon gas in the gap, providing the free energy and in particular the Casimir interaction of dielectric bodies with arbitrary shaped surfaces. A similar approach has been followed by Soltan et al. [16] for a dispersive medium between the bodies. Recently, Behunin and Hu applied path integrals to problems of the Casimir-Polder type (atom-surface interaction) in different nonequilibrium situations: in Ref. [17] is considered the Van der Waals interaction between an atom and a substrate in a stationary state out of global equilibrium; Ref. [18] is calculating atom-atom interactions in a quantized radiation field which is in a nonequilibrium state.

Our theoretical approach to the non-equilibrium Casimir interaction where the boundary conditions involve different, locally defined temperatures, is thus a synthesis between the Feynman path integral and the Keldysh-Schwinger formalism of nonequilibrium processes [19, 20].

The paper is organized as follows: after preliminary notations and introductions in Sec. 'Preliminaries', we introduce rather general expressions for retarded Green functions in the Casimir geometry of two planar boundaries (Sec. 'Retarded Green functions'). In Sec. 'Non-equilibrium action' is analyzed a Gaussian action in Schwinger-Keldysh space which yields the non-equilibrium correlations for the fields (Keldysh-Green functions). This action implements in particular the boundary conditions for the retarded Green functions. In Sec. 'The large distance limit' we discuss the retarded Green functions in the limit where the interfaces of the Casimir system are infinitely removed from each other. Using this limiting procedure, the Keldysh-Green function at any point in the gap is expressed via Keldysh functions of the single interface systems defined at the surfaces and via photon numbers in free space (Sec. 'Keldysh–Green functions'). As an application of the developed technique, we find in Sec. 'Example: sliding interfaces' the field correlation functions in a Casimir geometry with a sliding interface. This result is checked in Appendix D where the same problem is solved using Rytov fluctuation electrodynamics. We illustrate our results by analyzing the electromagnetic energy density between the plates in two typical non-equilibrium situations (Appendix E). Concluding remarks are given in Sec. 'Conclusions', while technical details are collected in the other Appendices.

2 Preliminaries

  1. Top of page
  2. Abstract
  3. 1 Introduction
  4. 2 Preliminaries
  5. 3 Retarded Green functions
  6. 4 Non-equilibrium action
  7. 5 The large distance limit
  8. 6 Keldysh–Green functions
  9. 7 Example: sliding interfaces
  10. 8 Conclusions
  11. Appendix A
  12. Appendix B
  13. Appendix C
  14. Appendix D
  15. Appendix E
  16. Acknowledgments
  17. References

2.1 Geometry of the problem

We consider two bodies with parallel and homogeneous boundaries located at inline image. The boundaries are in stationary conditions: their temperatures are constant in time, and their relative motion (if any) is uniform and parallel to each other. We can then assume that the EM field in the cavity [inline image] is stationary in time and homogenous in the inline image-plane. As a consequence, all relevant fields and correlation functions can be expanded in Fourier integrals with respect to frequency ω and wave vectors inline image along the interfaces. We use in the following the shorthand

  • display math(2.1)

For fields like the vector potential, we get a mixed representation

  • display math(2.2)

where the argument Ω is suppressed where no confusion is possible.

We work in the Dzyaloshinskii gauge inline image where due to the transversality condition for the electric field, the normal component inline image of the vector potential can be eliminated in favor of the tangential ones inline image. Furthermore, given the plane of incidence spanned by q and the normal to the interfaces, the dynamical variables of the EM field in the cavity are the following linear combinations

  • display math(2.3)

which are nothing but the vector potential amplitudes of s- and p-polarized waves. In the following, we call the components defined in Eq. (2.3) the Weyl representation of the vector potential or we say the vector potential is written in the Weyl basis [21]: inline image.

2.2 Weyl representation of free space Green function

The Green function (GF) of the EM field plays a crucial role in the following. Since it simply represents the vector potential due to a point current source, it is also represented by a mixed Fourier representation involving a tensor inline image (inline image). The latter is the solution of (given the gauge inline image) [22]

  • display math(2.4)

where (we set inline image)

  • display math(2.5)

In the following, we only need the tangential part of the GF inline image with inline image. (See Eqs. (3.16), (3.17), (3.18) below for the normal components.) Projecting both indices into the Weyl basis according to Eq. (2.3), we get from Eq. (2.4) the simple Helmholtz equation

  • display math(2.6)

where the 2 × 2 matrix inline image reads

  • display math(2.7)

The GF in free space (no boundary conditions) is written inline image. As is well known, it comes in two types: retarded inline image and advanced inline image Green functions. In the Weyl representation, they are given by

  • display math(2.8)

where

  • display math(2.9)

and the wave vector inline image is defined over the entire frequency axis by

  • display math(2.10)

The two cases corresponding to outgoing propagating and to evanescent waves, respectively. The infinitesimal imaginary part of inline image in (2.10) secures the analytical continuation of inline image to complex frequencies in the upper half plane. It entails the existence of the limit

  • display math(2.11)

for both propagating and evanescent waves. In addition, we have the symmetry relation

  • display math(2.12)

As a consequence, the retarded GF has the property inline image, as it should, being a response function between real-valued fields.

3 Retarded Green functions

  1. Top of page
  2. Abstract
  3. 1 Introduction
  4. 2 Preliminaries
  5. 3 Retarded Green functions
  6. 4 Non-equilibrium action
  7. 5 The large distance limit
  8. 6 Keldysh–Green functions
  9. 7 Example: sliding interfaces
  10. 8 Conclusions
  11. Appendix A
  12. Appendix B
  13. Appendix C
  14. Appendix D
  15. Appendix E
  16. Acknowledgments
  17. References

3.1 Single interface

We include stationary and translation-invariant (in the inline image-plane) boundary conditions in the retarded GF by adding reflected waves. Let us start with a single interface at inline image where the GF (subscript −) reads

  • display math(3.1)

for inline image. The first term is the same as in free space. The reflection matrix inline image at the lower interface takes a simple diagonal form in the frame where the lower body is at rest

  • display math(3.2)

whose matrix elements are for a

  • display math(3.3)

where inline image is the (dimensionless) impedance, and for a

  • display math(3.4)

where inline image is the dielectric permittivity and

  • display math(3.5)

the wave vector in the lower medium.

If only the upper medium is present, we have a GF similar to Eq. (3.1), for inline image

  • display math(3.6)

where inline image is the corresponding reflection matrix. We suppose here a generic form for inline image. The case of a sliding upper interface is discussed in Appendix D.

3.2 General properties

Let us collect a few general properties of the reflection matrices and the GFs. It follows from Eq. (D15), (D16), (D17), (D18) for inline image and from the diagonal form of inline image [Eq. (3.2)] that both fulfill the identity

  • display math(3.7)

The retarded GF (3.1) therefore satisfies

  • display math(3.8)

which is not the reciprocity condition because the wave vector inline image in the arguments of both sides is the same.

Taking into account Eqs. (2.12) and (D15), (D16), (D17), (D18), the reflection matrices in Eq. (3.2) and Sec. D satisfy the identity

  • display math(3.9)

where the asterisk denotes the element-wise complex conjugation. This entails that the symmetry relation is also valid for the retarded GF at a single interface

  • display math(3.10)

Below, we shall also deal with the advanced GF which is defined as

  • display math(3.11)

where the last equality follows from Eq. (3.8).

3.3 Planar cavity

The preceding properties carry over to the retarded and advanced GF in the cavity formed by two interfaces. By adding up multiply reflected waves, one finds the expression

  • display math(3.12)

where

  • display math(3.13)

and inline image are defined by swapping indices inline image in Eq. (3.13). The matrix inline image takes into account multiple reflections of photons in the cavity; it is given by the expression

  • display math(3.14)

where inline image is the unit matrix. An analogous formula gives inline image. Eq. (3.7) above gives us the property

  • display math(3.15)

which ensures that the generalized reciprocity relation (3.8) also holds for the cavity GF (3.12). Similarly, the relation (3.11) between retarded and advanced GFs remains true as well.

3.4 Normal components

To conclude this Section, we give the tensor components involving the normal direction. The following formulas can be shown from the wave equation Eq. (2.4) and the symmetry properties above (inline image and inline image):

  • display math(3.16)
  • display math(3.17)
  • display math(3.18)

3.5 Generalized impedance matrices

The reflection matrices inline image appearing in the expressions above are the solutions to a scattering problem at the planar interfaces. We discuss here an equivalent formulation in terms of generalized surface impedances. These will provide the link between boundary conditions imposed on fields and interactions with auxiliary fields restricted to the interfaces.

Evaluating the derivative with respect to z and inline image of the retarded GF (3.12) at the interfaces inline image, we come to the boundary conditions. With respect to the first coordinate z (derivative inline image), we find

  • display math(3.19)
  • display math(3.20)

where we defined the matrices

  • display math(3.21)

These generalize the concept of a surface admittance to a general reflection problem. Indeed, from the reflection amplitudes (3.3) for a metallic surface at rest, we get in the Weyl basis

  • display math(3.22)

Note that despite multiple reflections, the boundary conditions (3.19, 3.20) are of local character: they link the fields and their normal derivatives at the same position with the corresponding admittance matrices.

With respect to the second coordinate inline image (derivative inline image) of the GF, a similar calculation yields

  • display math(3.23)
  • display math(3.24)

where the admittance matrices appear transposed.

In the case of a single interface, we still find two boundary conditions at inline image. If the “missing” body is the upper one, for example, the admittance degenerates into inline image from Eq. (3.21). The boundary condition (3.20) then becomes equivalent to the Sommerfeld condition for an outgoing wave: inline image. This holds as long as inline image and in both the propagating and evanescent sectors.

3.6 Remarkable identity

Using the boundary conditions (3.19–3.24), their counterparts for the advanced GF inline image, and the Green equation (2.6), we come to the following property of the GFs in the two-plate geometry

  • display math(3.25)

In Eq. (3.25) we have introduced effective source strengths

  • display math(3.26)

which are easily shown to be antihermitian: inline image and symmetric: inline image Therefore, they have purely imaginary matrix elements: inline image Besides, using the parity properties (2.12) and (3.9) of inline image and the reflection matrices, the definition of the inline image entails

  • display math(3.27)

The identity (3.25) has a long history in macroscopic fluctuation electrodynamics. It has been noted by Eckhardt [23], although involving a spatial integral over volumes where the imaginary part of the permittivity is nonzero. The version we give here is technically somewhat simpler because only surface sources appear. This may be related to the “holographic principle” stating that under certain circumstances, all relevant properties of a (source-free) field are encoded in a hypersurface. This is obviously related to the classical Huyghens principle. An alternative proof of Eq. (3.25) is given in Appendix A using the Leontovich surface impedance boundary condition.

If one of the two interfaces is missing, an identity similar to Eq. (3.25) can be derived analogously. One simply has to replace the source strength for the missing interface by

  • display math(3.28)

where Θ is the unit step function. Note that only propagating waves appear on the free boundary: they represent the fields incident from infinity towards the interface.

4 Non-equilibrium action

  1. Top of page
  2. Abstract
  3. 1 Introduction
  4. 2 Preliminaries
  5. 3 Retarded Green functions
  6. 4 Non-equilibrium action
  7. 5 The large distance limit
  8. 6 Keldysh–Green functions
  9. 7 Example: sliding interfaces
  10. 8 Conclusions
  11. Appendix A
  12. Appendix B
  13. Appendix C
  14. Appendix D
  15. Appendix E
  16. Acknowledgments
  17. References

We now address the key problem of this paper: evaluate correlations for the EM field under non-equilibrium conditions. To this effect, we use the path integral method and work with an action for the EM field. An auxiliary field φ is introduced to enforce the boundary conditions at the two interfaces [8]. This technique was developed in previous work [9-11] for equilibrium situations. We extend the approach to the whole Schwinger-Keldysh space and define a Gaussian action for two coupled Bose fields: the vector potential A and the auxiliary field φ

  • display math(4.1)

The integration is over Ω and z. In the action (4.1), the vector potential inline image lives in Keldysh space and has two components that are called quantum inline image and classical inline image:

  • display math(4.2)

each of which contains the familiar transverse amplitudes in the Weyl basis

  • display math(4.3)

The 4 × 4 matrix inline image is the inverse of the Keldysh-Green (KG) matrix of the free EM field. In the Keldysh basis (4.2) for inline image, this KG matrix has the block structure

  • display math(4.4)

where inline image are the retarded and advanced Green functions for free space, introduced in Sec. 'Weyl representation of free space Green function'. The function inline image is the KGF for the free field, it collects symmetrized correlations of the vector potential. (A calculation is sketched in Sec. C below.) Our goal is to calculate its counterpart in the presence of the two interfaces that we denote inline image. It is given in the coordinate representation [24] as (inline image)

  • display math(4.5)

where inline image is the anti-commutator. We work here with the corresponding Fourier transforms with respect to inline image and to the tangential coordinates. In the Weyl basis, this gives the matrix inline image.

The auxiliary field inline image in Eq. (4.1) has eight components that are also grouped in the Keldysh structure

  • display math(4.6)

The components inline image themselves are

  • display math(4.7)

They are Weyl spinors localized in the lower (index −) and the upper (+) interface. Finally, the matrix inline image in the action (4.1) is related, as we shall see below, to the boundary conditions imposed at inline image.

4.1 Evaluating the path integral

We follow the standard path integral procedure and add source terms to the action (4.1)

  • display math

Then Gaussian path integrals are evaluated, first over the EM field and then over the auxiliary field. We get the generating functional for field correlations whose expansion to second order in inline image provides the following expression for the KG matrix:

  • display math(4.8)

where inline image is the solution of

  • display math(4.9)

The KG matrix inline image of Eq. (4.8) must have the block structure (4.4) with off-diagonal blocks that are Hermitian conjugates [Eq. (3.11)] and with an antihermitian Keldysh block

  • display math(4.10)

In addition, we impose the boundary conditions (3.19–3.24) on the off-diagonal elements inline image of (4.8). These conditions unambiguously define the structure of the matrices inline image and inline image in the action (4.1) as

  • display math(4.11)

where inline image is a 4 × 4 antihermitian matrix, and

  • display math(4.12)

Here the matrix inline image collects the boundary conditions at the two interfaces into a vector of operators in the Weyl basis

  • display math(4.13)

The boundary conditions (3.19–3.20) then take the integral form (common argument Ω suppressed)

  • display math(4.14)
  • display math(4.15)

with a derivative inline image acting to the left in the second line. For the advanced GF, the boundary conditions (3.23), (3.24) can be written as integrals over inline image and over inline image.

With the choices ((4.11), (4.12)), one can show that the Keldysh action (4.1) acquires that so-called “causal structure[19, 20].

Inserting the matrices inline image and inline image from Eqs. ((4.12), (4.11)) into Eq. (4.8), we find for the KG functions inline image the expressions

  • display math(4.16)
  • display math(4.17)
  • display math(4.18)

where inline image is a 4 × 4 matrix given by

  • display math(4.19)

The expressions (4.16) and (4.17) are integral forms of retarded and advanced GF because matrix products actually have to be read as the concatenation of integral operators. It is trivial to check that they satisfy the boundary conditions (4.14, 4.15). Using the explicit form (4.13) of inline image, we have checked in a straightforward calculation that Eq. (4.16) coincides with Eq. (3.12) for the retarded GF.

4.2 Distribution (Keldysh-Green) functions for photons

To analyse the expression (4.18) for the KG function, we first observe that the first term is equal to zero because inline image is a solution of the homogeneous equation corresponding to Eq. (2.6). (See Appendix B for details.) We shall argue that inline image that characterizes the auxiliary fields can be taken in block-diagonal form

  • display math(4.20)

where the quantity inline image [inline image] correspond to the lower [upper] boundary, respectively. Adopting this choice, a tedious, but elementary calculation leads us to (common argument Ω suppressed)

  • display math(4.21)

for the KG function. This has the same form as the Keldysh equation in quantum kinetics [4]. If we would take this formal analogy serious, then the quantities inline image would coincide with the Keldysh polarization operators on the boundaries (i.e., the KG correlation of interface polarization fields). In the non-equilibrium theory, these are nonlinear functionals of the Keldysh function for photons, so that in the semiclassical approximation, Eq. (4.21) would generate a kinetic equation of Boltzmann type for the photon distribution function.

In the two-plate geometry of the Casimir effect, however, this is not the case because the inline image are independent of the photon KG functions inline image, inline image, and inline image. They are rather determined by given macroscopic states of the bodies and their interfaces and act like a linear driving for the EM field in the cavity. This suggests an interpretation of Eq. (4.21) in the spirit of Rytov's fluctuation electrodynamics: the quantities inline image encapsulate the distribution of photon sources. They are the only piece of information needed to produce the (non-equilibrium) distribution function of photons in the cavity. For this reason, we call inline image the photon sources or fluctuating sources in the following.

The above interpretation also helps to understand the diagonal form for the source distribution function inline image in Eq. (4.20): the sources are located on the bodies' interfaces, and correlations between the macroscopic states of the two bodies are neglected. We intend to investigate corrections to this approximation in future work.

5 The large distance limit

  1. Top of page
  2. Abstract
  3. 1 Introduction
  4. 2 Preliminaries
  5. 3 Retarded Green functions
  6. 4 Non-equilibrium action
  7. 5 The large distance limit
  8. 6 Keldysh–Green functions
  9. 7 Example: sliding interfaces
  10. 8 Conclusions
  11. Appendix A
  12. Appendix B
  13. Appendix C
  14. Appendix D
  15. Appendix E
  16. Acknowledgments
  17. References

In this Section, we consider the limit inline image in order to fix the strength of photon sources by referring to the EM field in free space and near a single interface.

There are three different ways to take the limit:

  1. If we fix both points z, inline image in the cavity and go to the limit inline image, we recover Green functions in free space. In particular for the retarded Green function,
    • display math(5.1)
    with the free space Green function defined in Eq. (2.8).
  2. If the points stay at fixed distances inline image from the lower interface, we get
    • display math(5.2)
    where inline image is the Green function above a single interface. The notation inline image means that in the expression for inline image [Eq. (3.1)], a should be set to zero.
  3. Similarly, we may take points at positions inline image below the upper interface and get the Green function below a single interface:
    • display math(5.3)
    where the Green function inline image is defined similar to case (B) from Eq. (3.6).

We also need for the Keldysh-Green function Eq. (4.21) the limiting values when one position in the Green functions recedes to infinity. Keeping z fixed,

  • display math(5.4)
  • display math(5.5)

In this limit, the exponential inline image in the first line restricts the support to propagating waves (real inline image); it drops out when products with the advanced GF inline image are formed because of Eq. (2.8). In a similar way, we define for the single-interface limits (B, C) the functions

  • display math(5.6)
  • display math(5.7)

6 Keldysh–Green functions

  1. Top of page
  2. Abstract
  3. 1 Introduction
  4. 2 Preliminaries
  5. 3 Retarded Green functions
  6. 4 Non-equilibrium action
  7. 5 The large distance limit
  8. 6 Keldysh–Green functions
  9. 7 Example: sliding interfaces
  10. 8 Conclusions
  11. Appendix A
  12. Appendix B
  13. Appendix C
  14. Appendix D
  15. Appendix E
  16. Acknowledgments
  17. References

We now consider the KG function derived at Eq. (4.21) in the limit inline image in order to find the photon sources inline image. Using the notation inline image, inline image and inline image in the three limits (A), (B) and (C), we get

  • display math(6.1)
  • display math(6.2)

where the GFs on the right hand sides are defined in Eqs. (5.5), (5.6), (5.7) above. Manifestly, the sources in free space inline image are defined by the quantum state of free EM field. Similarly, the fluctuation sources on the physical interfaces [the lower one, inline image in case (B) and the upper one, inline image in case (C)] are defined by the macroscopic quantum state of the corresponding bodies.

We assume the statistical independence of the bodies and the free EM field [25] and come to the conclusion that the sources on the ‘free interfaces’ coincide:

  • display math(6.3)

and that the sources on the macroscopic interfaces are the same for one and for two plates:

  • display math(6.4)

By evaluating Eq. (6.2) at inline image, we can also express the interface sources in terms of the KG function there:

  • display math(6.5)

where inline image is the boundary value of the KG function for a system with a single interface

  • display math(6.6)

The free space photon sources inline image are calculated in Appendix C. In Sec. 'Example: sliding interfaces', we work out the quantities inline image for a specific example, allowing for the upper body to be in uniform motion relative to the lower one.

We are now ready to collect our main result. The explicit expressions for the GFs inline image [Eqs. (3.1, 3.11)] and for the photon sources inline image [Eq. (6.4)] are inserted into the KG function (4.21) to give

  • display math(6.7)

We find the amplitudes

  • display math(6.8)
  • display math(6.9)
  • display math(6.10)

where inline image, inline image, inline image are found by swapping the subscripts + and −, inline image has been defined in Eq. (3.14), and

  • display math(6.11)

These expressions give the distribution function of photons in a planar cavity (homogeneous along the inline image-directions and stationary in time) whatever the macroscopic state of the two boundaries, as encoded in inline image. Based on the assumption of statistical independence, the inline image are given by reference situations with a single interface [Eqs. (6.5, 6.6)].

In some cases (for instance in the Casimir effect), one needs to subtract the free-space KG function to get finite results for the relevant observables:

  • display math(6.12)

where inline image is defined by Eq. (6.1) with free-space photon sources inline image in the problem.

For completeness, the normal (z-) components of the KG functions are also given here. They are expressed in terms of tangential components, using the homogenous version of Eq. (2.4). We get in analogy to Eqs. (3.16), (3.17), (3.18) for the retarded GF

  • display math(6.13)
  • display math(6.14)
  • display math(6.15)

7 Example: sliding interfaces

  1. Top of page
  2. Abstract
  3. 1 Introduction
  4. 2 Preliminaries
  5. 3 Retarded Green functions
  6. 4 Non-equilibrium action
  7. 5 The large distance limit
  8. 6 Keldysh–Green functions
  9. 7 Example: sliding interfaces
  10. 8 Conclusions
  11. Appendix A
  12. Appendix B
  13. Appendix C
  14. Appendix D
  15. Appendix E
  16. Acknowledgments
  17. References

As an application of the theory developed so far, let us consider the KG function in the case of two bodies in relative motion. In this case in the limit (inline image) we have a sliding interface (moving parallel to x with velocity v). We suppose that we have equilibrium in the body's rest frame (with temperature inline image), similar to Refs.[26-28]. Using the Lorentz covariant formulation of the fluctuation-dissipation theorem [29], we get for the KG function in the limit (inline image) the expression (inline image)

  • display math(7.1)
  • display math(7.2)

where inline image is the Doppler-shifted frequency. A calculation starting from Eq. (6.5) yields the corresponding source located at the upper surface

  • display math(7.3)

where inline image is given in Eq. (3.26). The reflection matrix inline image that appears in inline image [see Eq. (5.7)] is calculated in Appendix D, Eqs. (D14), (D15), (D16), (D17), (D18), (D19). These expressions cover any relative velocities v.

The lower interface is in equilibrium at inline image. Therefore

  • display math(7.4)

and then using Eq. (6.5) we get for the sources there

  • display math(7.5)

Inserting Eqs. ((7.3), (7.5)) into Eq. (4.21), we come to the KG function for the two-plate geometry with a sliding upper interface

  • display math(7.6)

That the frequencies differ in the two inline image terms has been known in similar contexts since the pioneering work by Frank and Ginzburg on the Cherenkov effect (see Ref. [30] for an overview). This spoils any attempt to describe the sliding geometry by a global equilibrium assumption even if the two bodies are of the same temperature (as in Ref. [31]). The sign change between ω and inline image (anomalous Doppler effect) has also been noted in early work on field quantization in moving media, see, e.g., Jauch and Watson [32].

8 Conclusions

  1. Top of page
  2. Abstract
  3. 1 Introduction
  4. 2 Preliminaries
  5. 3 Retarded Green functions
  6. 4 Non-equilibrium action
  7. 5 The large distance limit
  8. 6 Keldysh–Green functions
  9. 7 Example: sliding interfaces
  10. 8 Conclusions
  11. Appendix A
  12. Appendix B
  13. Appendix C
  14. Appendix D
  15. Appendix E
  16. Acknowledgments
  17. References

We have calculated the photon distribution function between two parallel plates (Casimir geometry) under rather general conditions: on the two plates, any arbitrary stationary nonequilibrium state is allowed for. Using the Schwinger-Keldysh formalism of nonequilibrium field theory, we could express the Keldysh-Green function of photons at any point in the gap between the plates via the Keldysh functions of single interface systems, defined at the surfaces, and via photon numbers in free space. As a cross check of the results, we consider one plate sliding relative to the other in local equilibrium, and we find full coincidence of the results with Rytov theory, without any restriction in relative velocities.

Our approach is flexible enough to allow for the zero-temperature limit inline image to be taken. The example of the sliding plates of Sec. 'Example: sliding interfaces' then illustrates that one does not recover an equilibrium situation. The resulting frictional stress that opposes the relative motion is an interesting (and controversial [31, 33-35]) manifestation of an unstable vacuum state. For similar situations, we may quote the Klein paradox [36] (electron-positron pairs created by maintaining a static field configuration), and the Schwinger-Unruh effect (thermalization of an accelerated detector in vacuum). Indeed, one possible explanation for quantum friction involves the creation of particle pairs in the two plates [34, 35, 37], as pointed out in earlier work by Polevoi within the context of Rytov theory [26].

Finally, we suggest that the developed formalism is general enough to investigate generalizations beyond the standard fluctuation electrodynamics. The crucial assumptions of the latter are clearly spelled out: statistical independence of the sources localized on the macroscopic bodies. If this is relaxed, one has to establish the photon source strengths in some other way. Concepts from non-equilibrium kinetic theory like the Boltzmann equation or the balance of energy and entropy exchanges are likely to be instrumental here.

Appendix A

  1. Top of page
  2. Abstract
  3. 1 Introduction
  4. 2 Preliminaries
  5. 3 Retarded Green functions
  6. 4 Non-equilibrium action
  7. 5 The large distance limit
  8. 6 Keldysh–Green functions
  9. 7 Example: sliding interfaces
  10. 8 Conclusions
  11. Appendix A
  12. Appendix B
  13. Appendix C
  14. Appendix D
  15. Appendix E
  16. Acknowledgments
  17. References

Remarkable surface identity

In this Appendix, we give an alternative derivation of the remarkable identity (3.25) for the retarded and advanced Green functions. Coming back to its interpretation as the electric field radiated by a point electric dipole, we have the wave equation

  • display math(A1)

where the source dipole d is located at inline image. Multiply this equation by some vector field F, to be specified later, and integrate over a volume V with boundary A. Performing a partial integration leads to

  • display math(A2)

where n is the unit normal pointing into the volume V. We choose for the volume V the cavity bounded by two plates. Use one of the Maxwell equations and apply on the plates the surface impedance boundary condition inline image due to Leontovich [38], where inline image is the tangential electric field. This gives under the surface integral in Eq. (A2)

  • display math(A3)

More general boundary conditions (as for dielectrics) could be included by allowing for a q-dependent impedance. We now make the choice inline image (complex conjugate) and take the imaginary part of Eq. (A2). This removes the volume integrals over the real functions inline image and inline image. The rest leads to

  • display math(A4)

We recognize on the right-hand side the imaginary part of the Green function, inline image which can be written as the difference between retarded and advanced GFs. On the left-hand side, we recognize a source strength for surface currents given by the (positive) real part of the admittance inline image. This integral indeed represents the radiation by surface currents because the field inline image is related, by reciprocity [Eq. (3.8)], to the field generated at the vacuum point inline image by a source at the boundary point x.

Let us finally emphasize that Eq. (A4) greatly simplifies calculations in fluctuation electrodynamics because there is no need to perform a volume integration over sources distributed throughout the bulk of the bodies. It is sufficient to specify the radiation generated by the bodies on their boundaries. A technique that can be applied with a similar advantage is the generalized Kirchhoff principle where so-called ‘mixed losses’ are used to calculate correlation functions outside a body [39].

Appendix B

  1. Top of page
  2. Abstract
  3. 1 Introduction
  4. 2 Preliminaries
  5. 3 Retarded Green functions
  6. 4 Non-equilibrium action
  7. 5 The large distance limit
  8. 6 Keldysh–Green functions
  9. 7 Example: sliding interfaces
  10. 8 Conclusions
  11. Appendix A
  12. Appendix B
  13. Appendix C
  14. Appendix D
  15. Appendix E
  16. Acknowledgments
  17. References

Simplification of the KG function (4.18)

We show here that the first line of Eq. (4.18) for inline image vanishes:

  • display math(B1)

Recall that the KG function in free space inline image solves the homogeneous equation corresponding to Eq. (2.6). The dependence on the first argument z therefore reduces to inline image, inline image. We shall show that inline image acts like the unit operator on these kind of functions and cancels with the first term inline image in the left bracket of Eq. (B1).

This program can be carried out by straightforward algebra, calculating the matrix inline image by inversion from Eq. (4.19), and working out the action of the boundary operator inline image [Eq. (4.13)] on the exponentials:

  • display math(B2)

We present here a more compact proof whose starting point is the product inline image, i.e., the line vector

  • display math(B3)

By acting on this from the right with the block-diagonal, non-singular matrix

  • display math(B4)

we get

  • display math(B5)

The action of the operator inline image on these exponentials is precisely what we have to check. Using the operator representation of Eq. (B5), this is easily worked out to be

  • display math(B6)

where in step (*) the definition Eq. (4.19) was used. The same cancellation with the unit operator inline image in Eq. (B1) happens for the operator in the right bracket there which is just hermitean conjugate to this one.

Appendix C

  1. Top of page
  2. Abstract
  3. 1 Introduction
  4. 2 Preliminaries
  5. 3 Retarded Green functions
  6. 4 Non-equilibrium action
  7. 5 The large distance limit
  8. 6 Keldysh–Green functions
  9. 7 Example: sliding interfaces
  10. 8 Conclusions
  11. Appendix A
  12. Appendix B
  13. Appendix C
  14. Appendix D
  15. Appendix E
  16. Acknowledgments
  17. References

Free-space sources

To find the photon sources in free space inline image required for Eq. (6.5) we calculate the KG function by the standard mode expansion. The quantized vector potential is[36]

  • display math(C1)

where inline image and inline image are the familiar annihilation and creation operators for plane wave photon modes (wave vector k, polarization index λ). The normalized mode functions are (inline image)

  • display math(C2)

This is inserted into the definition (4.5) for the KG function inline image in the space-time domain. We take the tangential components, switch to the Fourier-Weyl representation and compare with Eq. (6.1). In this way, we find for the photon sources in free space

  • display math(C3)
  • display math(C4)

where we introduce two different matrices for left- and right-propagating photons

  • display math(C5)

which are, of course, diagonal in the polarization basis. They involve the Bose-Einstein distribution:

  • display math(C6)

in the simple case of a uniform temperature T. The photon sources in free space are thus defined by the average number of propagating photons that penetrate into the “cavity” through the corresponding “surfaces”.

Appendix D

  1. Top of page
  2. Abstract
  3. 1 Introduction
  4. 2 Preliminaries
  5. 3 Retarded Green functions
  6. 4 Non-equilibrium action
  7. 5 The large distance limit
  8. 6 Keldysh–Green functions
  9. 7 Example: sliding interfaces
  10. 8 Conclusions
  11. Appendix A
  12. Appendix B
  13. Appendix C
  14. Appendix D
  15. Appendix E
  16. Acknowledgments
  17. References

Rytov theory for a sliding interface

For comparison with the non-equilibrium Keldysh-Schwinger formalism, we outline here a calculation for the two-plate system within fluctuation electrodynamics, as developed by Rytov [40]. More details can be found in Refs.[26, 33, 37]. For simplicity, we construct fluctuating sources based on surface currents that are tangential to the two surfaces. As a concrete example, we consider a “sheared cavity”, i.e., the upper body is in relative motion with velocity v along the x-axis. Its reflection matrix inline image is found by applying the Lorentz transformation.

D.1 Spectral strength of surface currents

We begin by considering the surface currents of the sliding body in its rest frame inline image. From electric charge conservation, we get the charge density

  • display math(D1)

The currents in the laboratory frame K are found with the help of a Lorentz transformation. Using Eq. (D1) for the charge density inline image, we find in the Weyl basis the representation

  • display math(D2)

where the primed k-vector is given by the familiar Lorentz matrix (recall that inline image)

  • display math(D3)

For the matrix inline image in Eq. (D2), the calculation yields

  • display math(D4)
  • display math(D5)

We assume that the current density contains only surface current contributions inline image

  • display math(D6)

and get for the EM potential the source representation

  • display math(D7)

According to the framework of local Rytov theory[40], we define the commutator of surface currents as follows

  • display math(D8)

where the spectral strengths inline image (inline image, matrices in the Weyl representation) are to be fixed. The theory is local because currents “living” on different boundaries commute and are un-correlated—this is the main assumption in Rytov theory. To complete this definition, we calculate the commutator of the vector potential inline image which is nothing but the retarded GF [22]. Using the source representation (D7) and the source commutator (D8), we find (inline image omitted in all arguments)

  • display math(D9)

Now using the Kramers-Kronig relations for the retarded GF, we come to

  • display math(D10)

for the imaginary part. This equation coincides with (3.25) and therefore the spectral strengths inline image must be given by Eqs. (3.26).

The definition (D8) of inline image as a commutator of the surface currents yields their transformation law under a Lorentz transformation. Using Eq. (D2), we find the rule

  • display math(D11)

for the spectral strength on the upper (moving) interface (inline image). The primed quantities are evaluated in the frame co-moving with the body.

D.2 Local equilibrium spectra

We are now ready to compute the KG function according to its definition (4.5), and have to consider equilibrium averages of anticommutators for the surface currents. With the local equilibrium assumption, these are computed in the rest frames of the interfaces. For the lower interface, using the fluctuation–dissipation theorem [22] at temperature inline image, we get

  • display math(D12)

For the upper interface in its rest frame inline image, we have the same equation in terms of primed quantities, with the replacement inline image. Using transformation laws (D2) for the currents and (D11) for inline image, we get in the laboratory frame K

  • display math(D13)

Inserting Eqs. ((D12), (D13)) into the definition (4.5) of the KG function, we come to the result (7.6) found above within the Keldysh-Schwinger non-equilibrium framework.

D.3 Lorentz-transformed reflection matrix

From the transformation law (D11) for the surface current strength inline image, we can find the one for the reflection matrix inline image of the moving interface. Recalling Eq. (2.9) and the fact that inline image is an invariant for motions parallel to the inline image-plane, we get

  • display math(D14)

In the co-moving frame inline image, the matrix inline image is block-diagonal [see Eq. (3.2)] with elements inline image. The transformation law (D14) yields the following Weyl components

  • display math(D15)
  • display math(D16)
  • display math(D17)
  • display math(D18)

Here, the polarization mixing is governed by the angle θ with

  • display math(D19)

Appendix E

  1. Top of page
  2. Abstract
  3. 1 Introduction
  4. 2 Preliminaries
  5. 3 Retarded Green functions
  6. 4 Non-equilibrium action
  7. 5 The large distance limit
  8. 6 Keldysh–Green functions
  9. 7 Example: sliding interfaces
  10. 8 Conclusions
  11. Appendix A
  12. Appendix B
  13. Appendix C
  14. Appendix D
  15. Appendix E
  16. Acknowledgments
  17. References

Illustration: energy spectrum in the cavity

More details on the calculations in this Appendix can be found in Ref. [46].

E.1 Preparations

The average energy density inline image (in cgs units) is given by the following combination of correlation functions

  • display math(E1)

where inline image is a cartesian index. In the two-plate geometry considered in this paper, u depends only on the z-coordinate and has a natural spectral representation inline image in terms of frequency ω and parallel wave vector inline image).

Calculating the electric and magnetic fields from the vector potential, we end up with diagonal elements inline image, i.e. the two transverse polarizations,

  • display math(E2)

where the derivatives inline image are evaluated at inline image.

E.2 Simple limiting cases

The simplest reference situation is free space in global equilibrium at temperature T. The KG function inline image is given by Eq. (6.1) with the source spectra inline image [Eq. (C3)]. We then get

  • display math(E3)

which is even in ω because of Eq. (2.10) defining inline image. The step function reduces the spectral support to the light cone inline image with real inline image. It is easy to see that Eq. (E3) is equivalent to the Planck spectrum, including the zero-point energy.

As another check of Eq. (E2), we have considered a symmetrical cavity (identical plates at rest) at zero temperature where the energy density was calculated in Ref. [41]. After some straightforward algebra, the spectral expansion inline image (over real and positive frequencies) is identified, and complete agreement is found.

E.3 Two non-equilibrium examples

To illustrate the general case, we use Eq. (E2) and the representation (6.7) for inline image. In addition, we integrate the energy density over the cavity volume inline image in order to reduce the number of relevant parameters. The resulting spectrum inline image of the energy per area is dimensionless and plotted in the following as a function of frequency ω and wave vector q. We consider for illustration purposes two complementary situations: (a) two dielectric bodies with frequency-independent permittivity (index) inline image at zero temperature inline image, the upper one moving at velocity v along the x-axis. Situation (b) is taken in mechanical equilibrium (inline image) at two different temperatures inline image. One body is metallic, the other one dielectric as before.

Figure E1 illustrates the momentum distribution inline image of the energy spectrum in the q plane, at fixed frequency ω. The parameters of the bodies are given in the caption. By inspection of the formulas, we find that the surface sources have a spectral support inside the ‘polariton cone’ where the medium wave vector inline image is real [see Eq. (3.5)], i.e., inline image (orange circle). If the dielectric is moving, the border of the polariton cone, inline image, becomes an ellipse (left panel, red) that intersects the inline image axis at

  • display math(E4)

Above the Cherenkov threshold, i.e., inline image, the ellipse changes into two hyperbolas (right panel, red line). It is interesting that the simple setting of a dielectric in fast motion creates a situation quite similar to so-called hyperbolic or indefinite media. These have been studied recently; they show similar dispersion relations in the bulk and are approximately realized in meta-materials with an anisotropic dielectric response [42-45].

image

Figure E1. Spectrum of electromagnetic energy per area inline image between two dielectric bodies, one moving at velocity v along the x-axis, at a fixed frequency ω. Parameters: zero temperature, refractive index inline image (for both bodies in their respective rest frames, frequency-independent for simplicity), distance inline image free-space wavelengths. The contours give the dimensionless quantity inline image in steps of 0.1 between 0 and 1.3, higher values are clipped (white area), same scale in both panels. Dotted Green circle (radius inline image): free-space light cone; orange circle (radius inline image): propagating photons in the dielectric at rest (‘polariton cone'). Red ellipse (hyperbola): polariton cone in moving dielectric, as seen from the laboratory frame. (left panel) Velocity below Cherenkov threshold inline image: the polariton cone is an ellipse. (right panel) v above Cherenkov threshold: the polaritons of the moving dielectric fill two hyperboloids. A non-zero energy spectrum, but with less structure, is found on the other hyperboloid at inline image [Eq. (E4)].

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A setting with two temperatures is illustrated in Fig. E2: a hot dielectric facing a cold metal, both at rest. Here, cylindrical symmetry holds and the energy spectrum inline image depends only on the modulus q of the parallel wave vector. In the inline image-plane, one identifies the light and polariton cones (dotted green and orange), and the resonances of the planar cavity (red lines). The latter are quite weak because the dielectric plate is a poor reflector. The right panel in Fig. E2 shows broad peaks in the energy density at these resonances, as well as sharper features just inside the polariton cone (arrows). The spectrum differs from a global equilibrium situation (thin gray lines). This difference becomes small if inline image are close, as expected, but also for a highly conducting metal. The energy density is positive everywhere because we did not subtract the vacuum energy density (dashed blue line). The latter eventually dominates at large frequencies.

image

Figure E2. Energy spectrum between a hot dielectric and a cold metallic plate. (left panel) The contours give the dimensionless spectrum inline image in steps of 1 from 0 to 17. Red lines: modes in a perfectly reflecting cavity. Dotted green line: light cone inline image, orange line: polariton cone in the dielectric inline image. The blue arrows mark the cuts shown in the (right panel): energy density vs. frequency ω. Thick lines: two different temperatures (same as left panel), thin gray lines: global equilibrium at the average temperature inline image, dashed line: free-space spectrum at inline image and inline image. Parameters (for a reference temperature inline image): dielectric at inline image with index inline image, metal at inline image with impedance inline image such that the skin depth at inline image is inline image. We calculate the impedance from a Drude conductivity with relaxation time inline image. We have taken a relatively large distance inline image in order to push the cavity resonances (red lines) into the thermal spectral range (inline image).

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Acknowledgments

  1. Top of page
  2. Abstract
  3. 1 Introduction
  4. 2 Preliminaries
  5. 3 Retarded Green functions
  6. 4 Non-equilibrium action
  7. 5 The large distance limit
  8. 6 Keldysh–Green functions
  9. 7 Example: sliding interfaces
  10. 8 Conclusions
  11. Appendix A
  12. Appendix B
  13. Appendix C
  14. Appendix D
  15. Appendix E
  16. Acknowledgments
  17. References

One of us (V.E.M.) acknowledges financial support by the European Science Foundation (ESF) within the activity “New Trends and Applications of the Casimir Effect” (Exchange Grant 2847), and by the Deutsche Forschungsgemeinschaft (grant He-2849/4-1). V.E.M. thanks Prof. M. Kardar and M. Krüger for fruitful discussions and P. Milonni, G. V. Dedkov and A. A. Kyasov for comments on the manuscript. C.H. is indebted to G. Pieplow and H. Haakh for constructive criticism.

References

  1. Top of page
  2. Abstract
  3. 1 Introduction
  4. 2 Preliminaries
  5. 3 Retarded Green functions
  6. 4 Non-equilibrium action
  7. 5 The large distance limit
  8. 6 Keldysh–Green functions
  9. 7 Example: sliding interfaces
  10. 8 Conclusions
  11. Appendix A
  12. Appendix B
  13. Appendix C
  14. Appendix D
  15. Appendix E
  16. Acknowledgments
  17. References