A Boundary conditions and solutions for the vacuum contribution

# Casimir Force for Absorbing Media in an Open Quantum System Framework: Scalar Model

## Abstract

In this article we compute the Casimir force between two finite-width mirrors at finite temperature, working in a simplified model in dimensions. The mirrors, considered as dissipative media, are modeled by a continuous set of harmonic oscillators which in turn are coupled to an external environment at thermal equilibrium. The calculation of the Casimir force is performed in the framework of the theory of quantum open systems. It is shown that the Casimir interaction has two different contributions: the usual radiation pressure from vacuum, which is obtained for ideal mirrors without dissipation or losses, and a Langevin force associated with the noise induced by the interaction between dielectric atoms in the slabs and the thermal bath. Both contributions to the Casimir force are needed in order to reproduce the analogous of Lifshitz formula in dimensions. We also discuss the relation between the electromagnetic properties of the mirrors and the spectral density of the environment.

###### pacs:
03.70.+k; 03.65.Yz; 42.50.-p
\date

## I Introduction

Given the precision that has been recently achieved in the measurement of the Casimir forces (1), the use of realistic models for the description of the media that constitute the mirrors is an unavoidable step for the improvement of Casimir energy calculations, which is needed for comparison with the experimental data. Moreover, from a conceptual point of view, the theoretical calculations for mirrors with general electromagnetic properties, including absorption, is not a completely settled issue (2); (3); (4). Since dissipative effects imply the possibility of energy interchanges between different parts of the full system (mirrors, vacuum field and environment), the theory of quantum open systems (5) is the natural approach to clarify the role of dissipation in Casimir physics. Indeed, in this framework, dissipation and noise appears in the effective theory of the relevant degrees of freedom (the electromagnetic field) after integration of the matter and other environmental degrees of freedom.

Dielectric slabs are in general nonlinear, inhomogeneous, dispersive and also dissipative media. These aspects turn difficult the quantization of a field when all of them have to be taken into account simultaneously. There are different approaches to address this problem. On the one hand, one can use a phenomenological description based on the macroscopic electromagnetic properties of the materials. The quantization can be performed starting from the macroscopic Maxwell equations, and including noise terms to account for absorption (6). In this approach a canonical quantization scheme is not possible, unless one couples the electromagnetic field to a reservoir (see (3)), following the standard route to include dissipation in simple quantum mechanical systems. Another possibility is to establish a first-principles model in which the slabs are described through their microscopic degrees of freedom, which are coupled to the electromagnetic field. In this kind of models, losses are also incorporated by considering a thermal bath, to allow for the possibility of absorption of light. There is a large body of literature on the quantization of the electromagnetic field in dielectrics. Regarding microscopic models, the fully canonical quantization of the electromagnetic field in dispersive and lossy dielectrics has been performed by Huttner and Barnett (HB) (7). In the HB model, the electromagnetic field is coupled to matter (the polarization field), and the matter is coupled to a reservoir that is included into the model to describe the losses. In the context of the theory of quantum open systems, one can think the HB model as a composite system in which the relevant degrees of freedom belong to two subsystems (the electromagnetic field and the matter), and the matter degrees of freedom are in turn coupled to an environment (the thermal reservoir). The indirect coupling between the electromagnetic field and the thermal reservoir is responsible for the losses. As we will comment below, this will be our starting point to compute the Casimir force between absorbing media.

Regarding the Casimir force, the celebrated Lifshitz formula (8) describes the forces between dielectrics in terms of their macroscopic electromagnetic properties. The original derivation of this very general formula is based on a macroscopic approach, starting from stochastic Maxwell equations and using thermodynamical properties for the stochastic fields. As pointed out in several papers, the connection between this approach and an approach based on a fully quantized model is not completely clear. Moreover, some doubts have been raised about the applicability of the Lifshitz formula to lossy dielectrics (2); (3); (4).

The first calculation of the Casimir force between two absorbing slabs using a microscopic approach is, to our knowledge, due to Kupiszewska (9), who modeled dielectric atoms as a set of harmonic oscillators coupled to an environment with , in which the atoms can dissipate energy. In that work, a scalar field in 1+1 dimensions was considered, and all the environmental effects were described through a dissipative constant and a Langevin force. In the context of quantum open systems, this is tantamount to consider an Ohmic environment. The force between slabs was then obtained in terms of the reflection coefficients associated to the slabs, which are described by a complex dielectric function. This result was rederived using a Green-function method for quantizing the macroscopic field in absorbing systems in 1D in conjuction with scattering matrix approach (10). This was also extended to two identical absorbing superlattices (11). Esquivel-Sirvent et al demonstrated an alternative Green-function approach that makes the quantization of the field within the slabs unnecessary and calculate the Casimir force in an asymmetric configuration (12) which was earlier considered only in the lossless case (13). In Ref.(14), the Casimir force was calculated in a lossless dispersive layer of an otherwise absorbing multilayer by using the macroscopic field operators considered by (15). In the series of works (4); (16); (17), Rosa et al considered the evaluation of the electromagnetic energy density in the presence of an absorbing and dissipative dielectric, using the HB model for and constant dissipation. In particular, in the recent Ref.(17), they obtained the force density associated to spatial variations of the permittivity from which, in principle, one could obtain the Casimir force between slabs.

In this paper we will follow a program similar to that of Ref.(9), generalizing it by considering a general and well defined quantum open system. We will work with a simplified model analogous to the one of HB, assuming that the dielectric atoms in the slabs are quantum Brownian particles, and that they are subjected to fluctuations (noise) and dissipation, due to the coupling to an external thermal environment. We will keep generality in the type of spectral density to specify the bath to which the atoms are coupled, generalizing the constant dissipation model used in Ref.(9). Indeed, after integration of the environmental degrees of freedom, it will be possible to obtain the dissipation and noise kernels that modify the unitary equation of motion of the dielectric atoms. As we will see, general non-Ohmic environments do not provide constant dissipation coefficients in the equation of motion of the Brownian particles, even at high temperatures. Moreover, the spectral density of the environment determines the electromagnetic properties of the mirrors and therefore have a direct influence on the Casimir force.

In addition to the conceptual issues described above, there are additional motivations to consider detailed microscopic models of the Casimir force, in particular, the controversy about its temperature dependence. Assuming simple phenomenological descriptions of the materials based on the plasma or Drude models, the theoretical predictions for the Casimir force are different due to the contribution (or not) of the TE zero mode (18). At small distances such that , the differences are not too large, and the experimental results by Decca et al (19) seem to be well described by the plasma model. However, at large distances , the theoretical predictions differ by a factor . The Casimir force at such large distances have been recently measured (20), and the results are compatible with the Drude model after taking into account the interactions due to patch potentials on the surfaces of the conductors (some authors disagree with the evaluation of the effects of the patch potentials, see (21)). In any case, these controversies show that more detailed microscopic models are necessary to clarify the situation. For example, considering that the slabs contain classical or quantum non relativistic charges interacting via the static Coulomb potential, the result for the large distance limit agrees with that of the Drude model (22).

Another motivation for considering the Casimir forces in the framework of open quantum systems is the possibility of analyzing non equilibrium effects, such as the Casimir force between objects at different temperatures (23) and the power of heat transfer between them (24), including the time dependent evolution until reaching a stationary situation.

The paper is organized as follow: in next Section we shall present the model, the Heisenberg equations of motion for the different operators, and the vacuum and Langevin contributions to the field operator. In Section III we study much in deep the relationship between the microscopic model and the macroscopic electromagnetic properties of the mirrors. Section IV is dedicated to the evaluation of the Casimir force. After adding the vacuum and Langevin contributions, we show that the total force is given by a Lifshitz-like formula, where the reflection coefficients of the slabs depend on the properties of the atoms and the environment considered in the model. In Section V we comment on the relation of the quantum open systems approach developed in this paper and the Euclidean computation of the Casimir force. We summarize our findings in Section VI. The Appendices contain some details of the calculations.

## Ii The Model

### ii.1 The Lagrangian Density

With the aim of including effects of dissipation and noise in the calculation of Casimir force, we will use the theory of open quantum systems, having in mind the paradigmatic example of the quantum Brownian motion (QBM) (5).

The model consists of a system composed of two parts: a massless scalar field and dielectric slabs which, in turn, are described by their internal degrees of freedom (a set of harmonic oscillators). Both sub-systems conform a composite system which is coupled to a second set of harmonic oscillators, that plays the role of an external environment or thermal bath. For simplicity we will work in dimensions. In our toy model the massless field represents the electromagnetic field, and the first set of harmonic oscillators directly coupled to the scalar field represents the atoms in the slab.

Considering the usual interaction term between the electromagnetic field and the ordinary matter, the coupling between the field and the atoms in the slab will be taken as a current-type one, where the field couples to the velocity of the atoms. The coupling constant for this interaction is the electric charge . We will also assume that there is no direct coupling between the field and the thermal bath. The Lagrangian density is therefore given by:

 L = Lϕ+LS+Lϕ−S+LB+LS−B (1) = 12∂μϕ∂μϕ+4πη(12m˙r2(x;t)−12mω20r2(x;t))+4πηeϕ(x;t)˙r(x;t) + 4πη∑n(12mn˙q2n(x;t)−12mnω2nq2n(x;t))−4πη∑nλnqn(x;t)r(x;t),

where we have stressed the fact that and are also functions of the position, i.e. each atom interacts with a thermal bath placed at the same position. We have denoted by the density of atoms in both slabs. The constants are the coupling constants between the atoms and the bath oscillators. It is implicitly understood that Eq.(1) represents the Lagrangian density inside the plates, while outside the plates the Lagrangian is given by the free field one. The configuration of the slabs, of thickness and separated by a distance is shown in Fig.1.

The quantization of the theory is straightforward. It should be noted that the full Hilbert space of the model , where the quantization is performed, is not only the field Hilbert space (as is considered in others works where the field is the only relevant degree of freedom), but also includes the Hilbert spaces of the atoms and the bath oscillators , in such a way that . We will assume, as frequently done in the context of QBM, that for the three parts of the systems are uncorrelated and not interacting. Interactions are turned on at . Therefore, the initial conditions for the operators , must be given in terms of operators acting in each part of the Hilbert space. The interactions will make that initial operators to become operators over the whole space . The initial density matrix of the total system is of the form:

 ˆρ(0)=ˆρϕ(0)⊗ˆρS(0)⊗ˆρB. (2)

Since we are interested in the steady state (), we will assume thermal equilibrium at temperature between the three parts. Each density matrix in Eq.(2) will be taken as thermal type (we have set ).

### ii.2 Heisenberg equations of motion

Starting from the Lagrangian (1), it is easy to derive the Heisenberg equations of motion for the different operators. They are are given by:

 ˆpn=mn˙ˆqn, (3)
 ˙ˆpn=−mnω2nˆqn+λnˆr, (4)
 ˆp=m˙ˆr+eˆϕ, (5)
 ˙ˆp=−mω20ˆr+∑nλnˆqn, (6)
 □ˆϕ=4πηe˙ˆr, (7)

where the operators and are the conjugate momentum operators associated to the operators and respectively. Substituting Eq.(3) into Eq.(4), we get:

 mn¨ˆqn+mnω2nˆqn−λnˆr=0. (8)

As usual in the context of QBM, we solve the equations for the operators taking as a source, and replace the solutions into Eq.(6). In this way, the microscopic degrees of freedom in the slabs satisfy a Langevin-like equation of the form

 ˙ˆp=−mω20ˆr−mddt∫t0dτγ(t−τ)ˆr(x;τ)+ˆF(x;t), (9)

where the damping kernel and the stochastic force operator are the same as the ones of QBM (see (5) for a general and complete view). They are given by

 γ(t)=2m∫+∞0dωJ(ω)ωcos(ωt), (10)
 ˆF(t)=∑nλn√2mnωn(e−iωntˆbn+eiωntˆb†n). (11)

Here and are the annihilation and creation operators associated to , and is the spectral density that characterizes the environment. This function gives the number of oscillators in each frecuency for given values of the coupling constants :

 J(ω)=∑nλ2n2mnωnδ(ω−ωn). (12)

In order to obtain a true irreversible dynamics, we introduce a continuous distribution of bath modes, that replaces the spectral density by a smooth function of the frequency of the bath modes. Different functions will describe different types of environments. Physically, the thermal bath has a finite number of oscillators in a given range of frequencies. Then, a cutoff function must be introduced, containing some characteristic frecuency scale . Then, the spectral density takes the following form

 J(ω)=2πmγ0ω(ωΛ)α−1f(ωΛ). (13)

In the equation above, is the relaxation constant of the environment, while is the frequency cutoff function. The values of classify the different types of environments: corresponds to an ohmic environment (in which there is a dissipative term proportional to the velocity in the Langevin equation for the Brownian particle), while and describe subohmics and supraohmics environments respectively (5).

At equilibrium, the stochastic force operator in Eq.(11) and the damping kernel in Eq.(10) are not independent. The statistical properties of the stochastic force operator are given by the dissipation and noise kernels

 D(t−t′)≡i⟨[ˆF(t);ˆF(t′)]⟩=i[ˆF(t);ˆF(t′)]=−2∫+∞0dωJ(ω)sin[ω(t−t′)], (14)
 D1(t−t′)≡⟨{ˆF(t);ˆF(t′)}⟩=2∫+∞0dωJ(ω)coth(ω2T)cos[ω(t−t′)], (15)

which are the formal quantum open systems generalization of the relations employed in Ref.(9) for general environments and arbitrary temperature. Note that only the noise kernel involves the environmental temperature as a parameter. Considering Eq.(10) and Eq.(14), it is easy to show that:

 ddtγ(t−s)=−1mD(t−s), (16)

which relates the damping kernel to the statistical properties of the stochastic force operator .

All in all, the set of equations to solve now are Eqs.(5), (7), and (9).

It is possible to obtain a formal solution for the operators by considering the field as a source for the equation. This solution generalizes the crude approximation made in Ref. (9) for the evolution of the microscopic degrees of freedom in the mirrors. It is given by:

 ˆr(x;t)=G1(t)ˆr(x;0)+G2(t)˙ˆr(x;0)+1m∫t0dsG2(t−s)(ˆF(x;s)−e˙ˆϕ(x;s)), (17)

where are the Green functions associated to the QBM equation that satisfy:

 G1(0)=1,˙G1(0)=0, (18)
 G2(0)=0,˙G2(0)=1, (19)

for which, the Laplace transforms are given by

 ˜G1(z)=zz2+ω20+z˜γ(z), (20)
 ˜G2(z)=1z2+ω20+z˜γ(z), (21)

where is the Laplace transform of the damping kernel. Note that, given these conditions, one can prove that .

Inserting this solution into Eq.(7), we obtain the following equation for the field operator

 □ˆϕ+4πηe2m∫t0G1(t−τ)˙ˆϕ(x;τ)dτ=4πηe(˙G1(t)ˆr(x;0)+G1(t)˙ˆr(x;0)+1m∫t0G1(t−τ)ˆF(x;τ)dτ), (22)

subjected to the free field initial conditions:

 ˆϕ(x;0)=∫dk(1ωk)12(ˆakeikx+ˆa†ke−ikx), (23)
 ˙ˆϕ(x;0)=∫dk(1ωk)12(−iωkˆakeikx+iωkˆa†ke−ikx), (24)

where and are the annihilation and creation operators for the free field, and . The boundary conditions are the continuity of the field and its spatial derivative at the interface points.

We will compute the Casimir force from the -component of the energy-momentum tensor

 ˆTxx(x;t)=12((∂0ˆϕ)2+(∂xˆϕ)2). (25)

The force is explicitly given by

 FC=⟨ˆTextxx⟩−⟨ˆTintxx⟩, (26)

where the expectation values are taken on the regions outside the planes (Regions I or V) and between them (Region III), respectively, in a thermal equilibrium situation.

For this calculation, we need the explicit solution for the field equation (22). Considering the properties of the fundamental solutions , the first step is to take the Laplace transform of the equation in order to obtain

 ∂2∂x2ˆϕ(x;s) − s2(1+4πηe2m˜G2(s))ˆϕ(x;s)+sˆϕ(x;0)+˙ˆϕ(x;0)+4πηem˜G1(s)ˆF(x;s)+4πηem˜G1(s)ˆp(x;0) (27) +4πηeˆr(x;0)(s˜G1(s)−1)=0.

Since we are interested in the long-time behaviour (), we can omit terms containing the positions and momenta of the oscillators at . This assumption is well justified for , the scale associated with the damping or relaxation time of the environment. Therefore, the equation for the field can be approximated by:

 ∂2∂x2ˆϕ(x;s)−s2(1+4πηe2m˜G2(s))ˆϕ(x;s)=−sˆϕ(x;0)−˙ˆϕ(x;0)−4πηem˜G1(s)ˆF(x;s). (28)

This is the equation for the Laplace transform of the field, with the initial conditions and the Laplace transform of the stochastic Langevin force as sources. For simplicity, the spatial dependence of the matter terms was omitted, but it is important to remember that this expression is valid for the points inside the plates. Note also that, as there is no contribution from the atoms to the sources, the field operator acts on as an identity.

We propose a solution of the form , where two contributions are distinguished: the vacuum contribution , which results from the modified field modes; and the Langevin contribution , which depends linearly on the Langevin forces. Each contribution satisfies

 ∂2∂x2ˆϕV(x;s)−s2(1+4πηe2m˜G2(s))ˆϕV(x;s)=−sˆϕ(x;0)−˙ˆϕ(x;0), (29)
 ∂2∂x2ˆϕL(x;s)−s2(1+4πηe2m˜G2(s))ˆϕL(x;s)=−4πηem˜G1(s)ˆF(x;s). (30)

It is worth to note that the first equation presents only operators acting on , thus the associated field contribution is an operator on that space. In the same way, the second equation depends only on operators acting on , and therefore the Langevin contribution acts nontrivially only there. Taking all this into account the field operator reads

 Extra open brace or missing close brace (31)

In summary, the atoms act like a bridge between the field and the thermal bath, making no contributions to the total field. The problem, then, has been completely separated into two parts, that will be computed in the next subsection.

### ii.3 Vacuum and Langevin contributions

We will now solve the two independent Eqs. (29) and (30). For the vacuum contribution , the solution for is assumed to have the form:

 ˆϕV(x;t)=[∫+∞0+∫0−∞]dk2π(πωk)12(ˆake−iωktfk(x)+ˆa†keiωktf∗k(x)), (32)

where the first integral comprises the waves going from left to right, and the second one the waves going from right to left. The mode functions satisfy the equation:

 d2dx2fk(x)+ω2kn2(ωk)fk(x)=0. (33)

The refractive index given by

 n2(ωk)=1+4πηe2m˜G2(−iωk)=1+ω2Pω20−ω2k−iωk˜γ(−iωk), (34)

where is the plasma frecuency.

It is worthy to note that Eq.(33) is of the same form that the equation for the modes in a nonabsorbing dielectric medium (26), except that in this case the refractive index is frecuency-dependent. On the other hand, Eq.(34) depends on the Fourier transform of the damping kernel, which is associated to a general environment. There is no assumption neither about the spectral density of the environment nor on the value of the equilibrium temperature. In the next Section we will analyze in more detail the relation between the electromagnetic response of the medium and the spectral density.

The Eq. (33) for the modes can be solved by imposing continuity of the mode functions at all interfaces. The calculation is long but straightforward and is described in Appendix A.

In order to obtain the Langevin contribution, we must solve Eq. (30). Since the interaction begins at , the equations for the Laplace and Fourier transforms for this contribution are identical, and related by . Thus, the long-time solution can be written as , where satisfies

 ∂2∂x2ˆϕL(x;k)+k2ˆϕL(x;k)=0, (35)

in regions I, III and V; and,

 ∂2∂x2ˆϕL(x;k)+k2n2(k)ˆϕL(x;k)=−4πηemik(k20−k2−ik˜γ(−ik))ˆF(x;k), (36)

in regions II and IV. Here is the Fourier transform of the stochastic force operator. The explicit solution is presented in Appendix B.

In Section IV we will use the vacuum and Langevin contributions to the field operator in order to obtain the Casimir force between slabs. Before doing that, we will describe in more detail the relation between the macroscopic electromagnetic properties of the slabs and the microscopic model.

## Iii Generalized Permittivity from the quantum open system

With the aim of checking if our model is physically consistent, we will analyze the properties of the refraction index given in Eq.(34). Considering that is the permittivity of the material plates, we have:

 ϵ(ω)−1=ω2Pω20−ω2−iω˜γ(−iω). (37)

We can define the susceptibility kernel for the model as (25)

 χ(τ)=12π∫+∞−∞(ϵ(ω)−1)e−iωτdω=ω2P2π∫+∞−∞e−iωτω20−ω2−iω˜γ(−iω)dω. (38)

In principle, this integral can be evaluated by contour integration. Inversely, the permittivity can be expressed in terms of as:

 ϵ(ω)=1+∫+∞−∞χ(τ)eiωτdτ, (39)

which can be viewed as a representation of in the complex -plane. The permittivity is well defined when is finite for all and as , and its analytical properties can be studied directly from this expression.

All properties of the permittivity and susceptibility functions are strongly dependent on the Laplace transform of the damping kernel , which in turn depends on the spectral density of the environment. After taking the Laplace tranform of Eq.(10), we obtain

 ˜γ(s)=2m∫+∞0dωJ(ω)ωs(s2+ω2). (40)

As already described, a physical spectral density must incorporate a cutoff function. One could use a sharp cutoff or, alternatively, choose a continuous cutoff function that approaches zero rapidly for frequencies greater than the cutoff frequency , ensuring the convergence of the integral.

The first alternative, although simpler, makes the function not well-defined in the complex plane. The second alternative solves this problem and allows the use of the residue theorem to evaluate the integral. Inserting Eq.(13) into Eq.(40), we obtain

 ˜γ(s)=4γ0sπΛα−1∫+∞0dωωα−1(s2+ω2)f(ωΛ). (41)

In order to apply the residue theorem the integrand must be holomorphic on the superior complex half-plane, except in a finite number of points which are not on the real axis. Thus, different results are obtained considering distributions with or without poles on the superior half-plane. For an ohmic environment () and no cutoff function it is easy to see that .

In the case of distributions without poles (for example, a gaussian cutoff function), we have:

 ˜γNP(−ik)=πmkJ(k)≡˜γ1(k), (42)

where the subscript NP denotes the fact that the distribution has no poles. The resulting function is real and even in the variable .

On the other hand, the distributions usually considered in the literature have poles on (for example, a Lorentzian distribution). In these cases, for odd (with to mantain the convergence in Eq.(41)), we get

 ˜γP(−ik)=πmkJΛ(k)+i(−1)α−12πmk(−kΛ)α−1J−k(Λ), (43)

where the subscripts on denote the location of the pole. Although the resulting function is complex, the second equality in Eq.(42) remains valid.

Taking into account the above properties of the damping kernel, we now continue analyzing the properties of the permittivity and susceptiblility functions. As a particular example, in the Drude model one has . Therefore the denominator in Eq.(38) has two poles, both in the lower half -plane. Thus, as expected from a physical point of view, the susceptibility kernel shows a causal behavior, since it vanishes for . The analyticity of in the upper half -plane allows the use of Cauchy’s theorem, resulting in the well-known Kramers-Kronig relations for the real and imaginary part of the permittivity function .

In our more general case, the physical properties of are determined by the function . This dissipation function is given by the theory of quantum open systems through Eqs.(42) and (43), and depends on the chosen cutoff function.

Let us first consider, for simplicity, the case in which the cutoff function has no poles, represented by the Eq.(42). For a given spectral density, the denominator in Eq.(38) reads

 D(α)NP(ω)=ω20−ω2−i2γ0ω(ωΛ)α−1f(ωΛ). (44)

If we choose an ohmic environment () and no cutoff function (which is equivalent to put ), we reobtain the Drude model (if ) or the one-resonance model (when ). In principle, we could consider other values of while keeping . In this case, should be an odd function. For example, gives an ill-defined pole configuration, since one of the poles lies on the upper half -plane, breaking the analyticity of the integrand in Eq.(38), and resulting in a non-causal susceptibility, which turns out to be unphysical.

Therefore we see that for this supraohmic environment the use of a cutoff is unavoidable. We may use an analytical cutoff (like a gaussian function), or a lorentzian cutoff function. The first alternative leads to a denominator whose zeroes cannot be obtained analytically. The second alternative, valid for such that is an odd function, leads to a denominator

 D(α)P(ω)=(Λ2+ω2)(ω20−ω2)+2γ0Λ3−αω2((−1)α−12Λα−2−iωα−2)(Λ2+ω2), (45)

which for gives

 D(3)P(ω)=Λω20−iω20ω−(2γ0+Λ)ω2+iω3Λ−iω. (46)

We denote the zeroes of as (with ). The three roots turn out to be located in the lower half -plane, which ensures the causality property. Also, one of the roots is purely imaginary () and the others two have the same (negative) imaginary part but opposite real parts (). Thus, the susceptibility kernel reads

 χ(3)P(τ)=−(ωPω0)2[(Λ−ω0|x1|)(x1−x2)(x1+x∗2)e−ω0|x1|τ+2Re[(Λ−iω0x2)(x2−x1)(x2+x∗2)e−iω0x2τ]]θ(τ), (47)

where it is clear that it is a causal real function and, due to the negativity of the imaginary part of the roots , we have for as is expected. We also have that but , and therefore the asymptotic expression found in (25) still remains valid, as well as the Kramers-Kronig relations.

It is worth noting that an ohmic environment () can be also studied with a cutoff function, obtaining similar results.

All in all, we have shown that our model is physically consistent and generalizes previous results for the permittivity of absorbing media, including as particular case the Drude model. Plasma-like models do not contain dissipation and can be obtained by taking , that corresponds to no coupling between the system and the bath.

## Iv Casimir Force

### iv.1 The Energy-Momentum Tensor and the Different Contributions to the Casimir Force

Once determined the two contributions to the field, we proceed to compute the Casimir force between the plates as given by Eq. (26). For this purpose, it is necessary to compute the expectation value of which is given by Eq. (25).

Considering the vacuum and Langevin contributions according to Eq.(31), we have

 ˆTxx(x;t) = 12((∂0(ˆϕV+ˆϕL))2+(∂x(ˆϕV+ˆϕL))2)=ˆTVxx⊗IS⊗IB+Iϕ⊗IS⊗ˆTLxx+(∂xˆϕV)⊗IS⊗(∂xˆϕL) (48) +(∂tˆϕV)⊗IS⊗(∂tˆϕL).

It is worth to remark that there are cross terms which act over two parts of the total Hilbert space.

As we are interested in the steady state of the system, which is assumed at thermal equilibrium, each part of the total density matrix is represented by a thermal-type density matrix. On the other hand, both field contributions and are linear on the annihilation and creation operators of their respective parts of the total Hilbert space. Thus, the cross terms do not contribute to the force in the case of thermal equilibrium. Then, the problem is reduced to compute the expectation values over thermal states of the operators and , i.e.,

 f=⟨ˆTxx⟩=Trϕ(ˆρϕˆTVxx)+TrB(ˆρBˆTLxx)=fV+fL. (49)

Thus, the Casimir force also have two contributions:

 Missing \left or extra \right (50)

### iv.2 Vacuum Casimir Force

For the vacuum contribution, is quadratic in the annihilation and creation operators. Thus, in order compute the expectation value over the thermal state, we need to evaluate the expectation values of the products of the annihilation and creation operators. These are given by the known expressions:

 ⟨ˆakˆak′⟩=⟨ˆa†kˆa†k′⟩=0, (51)
 ⟨ˆakˆa†k′⟩=δ(k−k′)(1+N(ωk)), (52)
 ⟨ˆa†kˆak′⟩=δ(k−k′)N(ωk), (53)

where .

Taking into account Eq.(32), we have, in region

 fVl(x)=Trϕ(ˆρϕˆTV,lxx)=14[∫+∞0+∫0−∞]dk2πcoth(βωk2)⎛⎝ωk|flk(x)|2+1ωk∣∣ ∣∣dflkdx∣∣ ∣∣2⎞⎠, (54)

which is identical to the expression for a non-absorbing medium except that, in this case, there is a thermal factor related to the temperature of the field.

Using the solutions for the modes functions in regions I and III (see Appendix A), the vacuum Casimir force is given by

 FVC=fVI−fVIII=12∫+∞0dk2πkcoth(βωk2)[1+|Rk|2+|Tk|2−2(|Ck|2+|Dk|2)]. (55)

The coefficients , , , and are explicitly given in Appendix A. It is worth noting the appearance of a thermal global factor in the last expression, which comes from the field’s thermal state at temperature , based on the equilibrium assumption.

### iv.3 Langevin Casimir Force

For computing the Langevin contribution to the force, it is necessary to know the expectation value, over the bath’s thermal state, of the force operator, where binary products are evaluated at different frequencies.

For any time-dependent hermitian operator, the expectation value evaluated at different times, corresponds to the correlation function of the operator. This matches with one-half of the anti-commutator expectation value at different times. Thus, making Fourier transforms over both times, we can compute the desired products of the Fourier transform of the force operator, at different frequencies.

For the case of thermal equilibrium, the anti-commutator expectation value at different times of the force operator is provided by the QBM theory. One can show that it matches the noise kernel of Eq.(15). Thus, we obtain

 ⟨{ˆF(k);ˆF(k′)}⟩=J(ωk)coth(βωk2)δ(k+k′). (56)

Considering that in our case the stochastic force operator depends on the position where the atom is located, we finally have

 ⟨ˆF(x;k)ˆF(x′;k′)⟩=δ(x−x′)J(ωk)2ηcoth(βωk2)δ(k+k′), (57)

where we have included the atom density due to dimensional issues. As it can be seen, the frequency spectrum is not flat.

Taking all this into account, in regions I and III are given by:

 ˆTL,Ixx(x;t)=∫+∞−∞dk2π∫+∞−∞dk′2π(−kk′)ˆW1(k)ˆW1(k′)e−i(k+k′)(x+t), (58)
 ˆTL,IIIxx(x;t)=∫+∞−∞dk2π∫+∞−∞dk′2π(−kk′)e−i(k+k′)t(ˆW2(k)ˆW2(k′)ei(k+k′)x+ˆW3(k)ˆW3(k′)e−i(k+k′)x). (59)

The coefficients in these equations are given in Appendix B, and are linear functions of the Fourier transform of the stochastic force operator. Taking into account the explicit expressions in Appendix B and Eq.(57), the desired expectation values are:

 ⟨ˆW1(k)ˆW1(k′)⟩ = Missing or unrecognized delimiter for \right (60) × [n1(1−e−z2)(|t|2ez2+|rn+rei2ka|2+ez2|1+rrnei2ka|2+|t|2|rn|2) + 2n2Re(|t|2i(e−iz1−1)r∗n+i(1−eiz1)(1+r∗r∗ne−i2ka)(rn+rei2ka))],
 ⟨ˆW2(k)ˆW2(k′)⟩ = Missing or unrecognized delimiter for \right (61) × [n1(1−e−z2)(1+|rn|2e−z2)+2n2e−z2Re(i(1−eiz1)rn)],

where , i.e., and , , and . The explicit expressions for the coefficients and can be found in Appendix A.

Therefore, the Langevin contribution to the force in regions I and III are given by

 fLI = TrB(ˆρBˆTL,Ixx)=∫+∞−∞dk2π |W(k)|2 2πmJ(ωk)˜γ1(k) e−z2coth(βωk2) (62) × [n1(1−e−z2)(|t|2ez2+|rn+rei2ka|2+ez2|1+rrnei2ka|2+|t|2|rn|2) + 2n2Re(|t|2i(e−iz1−1)r∗n+i(1−eiz1)(1+r∗r∗ne−i2ka)(rn+rei2ka))],
 fLIII = TrB(ˆρBˆTL,IIIxx)=∫+∞−∞dk2π |W(k)|2 4πmJ(ωk)˜γ1(k) coth(βωk2)(1+|r|2) (63) × [n1(1−e−z2)(1+|rn|2e−z2)+2n2e−z2Re(i(1−eiz1)rn)].

Taking advantage that the integration is over all the values of , the fact that the change is equivalent to complex conjugation, and the second equality of Eq. (42), we obtain

 fLI=∫+∞0dk2πk2|n+1|2|n|28|t|2