Born-series approach to the calculation of Casimir forces
The Casimir force between two objects is notoriously difficult to calculate in anything other than parallel-plate geometries due to its non-additive nature. This means that for more complicated, realistic geometries one usually has to resort to approaches such as making the crude proximity force approximation (PFA). Another issue with calculation of Casimir forces in real-world situations (such as with realistic materials) is that there are continuing doubts about the status of Lifshitz’s original treatment as a true quantum theory. Here we demonstrate an alternative approach to calculation of Casimir forces for arbitrary geometries which sidesteps both these problems. Our calculations are based upon a Born expansion of the Green’s function of the quantised electromagnetic vacuum field, interpreted as multiple scattering, with the relevant coupling strength being the difference in the dielectric functions of the various materials involved. This allows one to consider arbitrary geometries in single or multiple scattering simply by integrating over the desired shape, meaning that extension beyond the PFA is trivial. This work is mostly dedicated to illustration of the method by reproduction of known parallel-slab results – a process that turns out to be non-trivial and provides several useful insights. We also present a short example of calculation of the Casimir energy for a more complicated geometry, namely that of two finite slabs.
The existence of an attractive force between two conducting plates, known as the Casimir effect Casimir (1948), is one of the most striking predictions of quantum field theory. Advances in experimental techniques at the nano-scale have meant that the Casimir force has been attracting increasing attention as a phenomenon which may be harnessed in micro and nano-mechanical systems (MEMS/NEMS). Continuing applications include a wide variety of investigations into the forces at work in nanoscale devices relevant to emerging quantum technologies Neto et al. (2005); Chan et al. (2001); Munday et al. (2006); Capasso et al. (2007); Rodriguez et al. (2011); Kats (1977); Gies and Klingmüller (2006).
These applications have led workers in the field to go far beyond the idealised parallel-plate geometry first considered by Casimir, leading to a number of theories ranging from techniques based directly on Lifshitz’s original theory of fluctuations in media Landau et al. (1960); Golestanian (2009), to scattering approaches Marc Thierry Jaekel and Serge Reynaud (1991); Rahi et al. (2009), to geometric optics Jaffe and Scardicchio (2004) (for comprehensive reviews, see Dalvit et al. (2011); Bordag et al. (2009)). These extensions are fraught with difficulties, however, particularly in non-trivial geometries. These problems arise chiefly from the fact that the Casimir force is non-additive, meaning that one cannot reliably approximate the Casimir force for a complex geometry by considering it to be composed of multiple simpler geometries, as is often done across physics and engineering (e.g. finite-element analysis). A scheme known as the proximity force approximation (PFA) attempts to adapt the philosophy of finite-element analysis to Casimir forces, but has been shown many times to be significantly in error when compared to exact results (see, for example, Maia Neto et al. (2008); Rodriguez et al. (2007)).
Another contemporary issue in Casimir physics is much of the literature’s reliance upon the theory of intermolecular interactions between extended bodies developed in Landau et al. (1960), often referred to simply as the Lifshitz theory (see footnote
An alternative approach to calculation of quantum electrodynamical effects in material bodies is known as macroscopic QED Gruner and Welsch (1996) and is based on the introduction of a ‘noise current’ source term in Maxwell’s equations in order to satisfy the fluctuation-disspation theorem and preserve the canonical commutation relations of the electromagnetic field operators. It is closely related to the theory of Huttner and Barnett Huttner and Barnett (1992) where the medium is represented by various interacting matter fields. While extremely elegant and powerful, the model of Huttner and Barnett is somewhat difficult to apply to in homogenous media (see, for example, Eberlein and Zietal (2012); Yeung and Gustafson (1996)). For this reason we choose to work with the noise-current theory of Gruner and Welsch (1996), which encodes the effects of an inhomogenous medium via its electromagnetic dyadic Green’s function – a quantity which is well-known for several systems of interest. This quality has meant that the noise-current theory has found considerable success in terms of its power and applicability to a wide range of problems (see, for example, Buhmann et al. (2005); Tomaš (2006); Khanbekyan et al. (2008); Buhmann and Welsch (2006)), but in its original form the theory does not attempt a rigorous quantisation (unlike that of Huttner and Barnett), which is a obviously a desirable property for any quantum theory. This suggests that macroscopic QED may have similar problems as Lifshitz’s original theory. However, macroscopic QED in the form presented by Gruner and Welsch (1996) has relatively recently been put on a firm canonical foundation Philbin (2010, 2011), in which the noise source term appears naturally. A useful practical consequence of the successful canonical quantisation of Gruner and Welsch (1996)’s treatment of macroscopic QED is the elimination of some uncertainties Philbin et al. (2010) in the correct form of the Casimir energy. This means that for arguably the first time workers in the field are able to confidently use the techniques and results of the noise-current approach to macroscopic QED to investigate the Casimir effect, safe in the knowledge that the theory rests on a canonical foundation.
In this work we will demonstrate an approach to calculating the Casimir force between objects of arbitrary shape that sidesteps the above problems concerning the possible inadequacies of the PFA and Lifshitz’s original theory. To do so, we adapt an approach known as dielectric-contrast perturbation theory, where the dyadic Green’s function that describes the electromagnetic field subject to the boundary conditions imposed by a set of objects is approximated via a Born series Buhmann and Welsch (2006). We can then use these approximate Green’s functions directly in the formulae presented in Philbin (2011), meaning our treatment is entirely canonical and independent of Lifshitz’s original theory. Furthermore, it will be shown the shape of the objects enters into the calculation as the limits on a volume integral, which can, in principle, be freely chosen. This means that our approach is free from the problems associated with the parallel-plate foundations of the PFA. Arbitrary-order terms in the Born series can be included in a systematic way, this fact turns out to be vital for any calculation of the Casimir effect.
The assumption that the Born series approach rests upon is that the dielectric function of an arbitrary arrangement of objects is sufficiently similar to the dielectric function for some simpler arrangement in which we can make exact calculations, as shown schematically in fig. 1.
We can model the arbitrarily-shaped material body (fig. 1a) as the sum of a simpler arrangement (1b) for which is known, and a (1c) for which is not known. The latter can be of arbitrary shape and is considered as a perturbation to the former, and through the machinery of the Born series it can be taken into account to arbitrary precision by including the required number terms. In practice, however, one requires that the ‘extra’ part of the geometry (fig. 1c) is ‘small’
By far the most significant advance is that this work is based entirely on the canonically-derived Casimir force and energy expressions found in Philbin (2011). Consequently, this work can be viewed as the first canonical treatment of the dielectric-contrast approach to the Casimir effect, in contrast to Golestanian (2009) which is based on the archetypal Lifshitz theory.
Unlike Golestanian (2009), we do not restrict ourselves to vacuum for the medium between the dielectric bodies. This is a significant advance since the theory is based upon there being only a small difference in dielectric function of the collection of material bodies and that of the intervening medium.
We use a spectral rather than spatial representation of the unperturbed Green’s function. While this may seem like a minor technical difference, use of the spectral representation means that our calculation is relatively easily generalizable to more complex unperturbed Green’s functions than the homogenous medium we shall use, namely cylinder, sphere and layered versions thereof.
In addition to these, a pedagogical difference is that instead of calculating the Casimir energy directly as is done in Golestanian (2009), we systematically piece together a calculation of the Casimir energy, which turns out to mean that we get several useful results and insights along the way.
The structure of this work is as follows. We will begin by reproducing known results for the Casimir force between two parallel, infinite dielectric slabs in section II. We shall see that in order to generalise to finding the force for more complex geometries one actually requires the Casimir energy, so in section III we will also reproduce some known Casimir force results via calculation of the Casimir energy. Finally, in section IV we will present a short numerical example where we investigate the Casimir energy density between finite dielectric slabs.
Ii Casimir Force
ii.1 Basic expressions
For dielectric bodies which are translationally invariant in two out of three spatial dimensions, the Casimir force can be expressed entirely in terms of the so-called scattering dyadic Green’s function which describes the geometry-induced modification of the electromagnetic field in the region between the dielectric bodies Raabe and Welsch (2006); Philbin (2011). The component of the Casimir stress tensor in Cartesian co-ordinates in a region with position and frequency-dependent permittivity and permeability
with the following definitions
with denoting curl with respect to the second index of the tensor , so that (no minus sign) for a vector . Eq. (1) holds for arbitrary media obeying the Kramers-Kronig relations, and its nature as an integral over complex frequencies means it automatically includes the various contributions from different types of mode (travelling, evanescent, surface plasmon, etc, as explicitly shown for related calculations in Bennett and Eberlein (2012a, b, 2013)). The diagonal components of the Casimir stress tensor directly delivers the force per unit area between two objects, so the problem is essentially reduced to finding the dyadic Green’s function of the electromagnetic field for that particular geometry. This is only analytically possible for spheres, infinite planes and cylinders or layered versions thereof since the Helmholtz equation is separable only in those geometries
However, as shown by Buhmann and Welsch (2006) the dyadic Green’s function for an arbitrary geometry may be expanded in a Born (Dyson) series. The defining equation for the whole dyadic Green’s function (not just its scattering part) is
where is the unit dyadic . As shown in Buhmann and Welsch (2006), this may be expanded in a Born series about some known ‘background’ Green’s function
where is the difference between the entire dielectric function and that of the background material. If we now specify that the system at hand is an object of dielectric function described by some volume sitting in some ‘background’ dielectric material i.e.:
we can restrict the integrals to being over the volume , and also bring the dielectric functions outside the integrals
The above equation is exact, and holds for any volume . However, from here on we take the unperturbed Green’s function to be that for a homogenous medium, which we shall call ;
where is a functional that extracts the homogenous part of its argument. Using Eq. (10) we may rewrite this as
The first line of Eq. (13) vanishes by definition, leaving us with
We emphasise the fact that has, in general, a homogenous part that must be subtracted, in contrast to refs Golestanian (2009) and Buhmann and Welsch (2006). As discussed in detail in Appendix A, this complication arises at spatial points where the homogenous part of the dielectric function is not equal to the unperturbed dielectric function. For example, if the unperturbed dielectric function is that for vacuum, and the Green’s function is calculated inside an object then the homogenous part of the Green’s function is that which would arise if the object were of infinite extent which. This means it is different from the unperturbed Green’s function, which would be that for vacuum. This type of situation is not considered in Golestanian (2009) or Buhmann and Welsch (2006), but its inclusion is necessary here as we shall see in section III.
ii.2 Calculation of the force
For our purposes the most convenient form of the homogenous Green’s function is a Fourier-space decomposition into vector wave functions Chew (1995)
with , and and . The functions and are:
with , where is a unit vector perpendicular to . The magnitude of the wave vector is given in terms of the frequency by . The matrix is given by:
The Green’s functions for more complicated unperturbed geometries (cylinder, sphere or layered versions thereof) can also be written as products of vector wave-functions in an identical way to (15). This is the reason behind our statement in section I that the spectral form of the Green’s function is more appropriate to generalisation of our calculation to more complex unperturbed geometries than the homogenous medium presented here.
The Casimir geometry we will investigate is shown in Fig. 2.
We will take the unperturbed dielectric function to be that of the middle slab so that . Writing , we have that the Casimir-geometry volume over which we shall integrate is given by
with . To find the force we require the component of the Casimir stress tensor (1) in the region between the plates, which depends only on the Green’s function between the plates, so we may let in Eq. (1), giving:
We note that and are independent of so each successive order of approximation of about is simply given by plugging each successive term in the Born series of and into Eq. (19). This will not be so simple in the calculation of the Casimir energy due to the requirement to work inside the slabs as well as in the gap between them.
To find the Casimir force between the two slabs we need the Green’s function given by Eq. (14) in the region between the plates only. This region has the special property of the unperturbed Green’s function being equal to the homogenous part of the whole Green’s function. As shown in appendix A, this means that the following must hold;
leaving us with
where we have used the obvious notation
Equation (25) makes physical sense because if the unperturbed Green’s function is equal to the homogenous part of the whole Green’s function then the scattering Green’s function coincides with the perturbation to that homogenous part. Again we emphasize that this is not true in general, it is specifically not true in a region where the unperturbed Green’s function is different from the homogenous part of the whole Green’s function, as discussed in detail in appendix A.
To find the contribution to the Green’s function between the slabs that is first-order in we need to evaluate the term of Eq. (25). This is given by
Since runs only over the volume and we are calculating in the region , we may immediately ignore the function part of shown in Eq. (15). Denoting the parallel frequency integration variable contained within one of the factors in the integrand of Eq. (27) as , one easily sees that the integral trivially evaluates to functions over and , meaning that the integral is also trivial. Going into polar co-ordinates defined by and and carrying out the angular integral , we find for the first-order terms and in the approximations of and [Eqs. (22)]:
with , and where we have defined
The quantities are matrices given by
The choice of notation reminds the reader that these quantities are specific to the middle of the geometry (the region between the slabs), later on we shall consider more general versions of these matrix elements. We note that the matrix elements pertaining to the magnetic-type terms are not obtainable from the electric-type matrix elements through a multiplicative factor, as one might expect from a duality relation Buhmann and Scheel (2009). The reasons for this are discussed in appendix B. The integral in Eq. (28) is elementary,
Using this in Eq. (28), we find agreement with the linear term in the Taylor expansion of the exact Green’s function Tomaš (2002) for with between the slabs
so we conclude that the Casimir stress tensor (and consequently the force) vanishes to linear order in the dielectric contrast ;
This result makes physical sense because the first term in the Born series only ‘knows’ about one plate since it contains only a single scattering event, as shown in fig. 3. This means there cannot be a Casimir force to this order since the attractive force between the objects results from their interaction with each other, mediated by the electromagnetic field. Clearly this interaction cannot take place if there is only one scattering event.
The second-order Green’s function in the region between the plates is given by the term of Eq. (25).
As explained below Eq. (27), in the first-order calculation the restriction , equivalent to , meant we could ignore the delta function part of the homogenous Green’s functions and entering into Eq. (27). However, in the second-order calculation we have an additional factor of , meaning we have a term proportional to . This does not vanish since the and integrals run over the same region as each other. Because of this we rewrite the homogenous Green’s function as the sum of a delta function part (contributing at equal spatial points only), and a ‘propagating’ part which contributes for all spatial points:
Since , we have for the two outer factors in the second line of (35)
but for the middle factor:
This means the second-order equal-point Green’s function naturally splits into two parts,
Though we shall not repeat the algebra here, the above analysis for also holds for since the plane-wave nature of the homogenous Green’s function means that the derivative operators can only generate various overall factors of or . Proceeding, we substitute Eq. (37) into (44), make the same integral manipulations that took Eq. (27) to (28) and evaluate the integral over , giving
where , and
[c.f. Eqs. (21)]. The matrices are defined as:
In an identical way to section II.2.1, we then note that the Casimir force arising from this term is proportional to
So, as expected, the single-scattering part of the second-order Green’s function [given by Eq. (43)] does not contribute to the Casimir force. We are then left with the double-scattering part only, given by Eq. (42). The full integral obtained by substituting Eq. (37) into (42) is far too cumbersome to write explicitly here, but it is worth noting that its integrand is proportional to
which results in different behaviour depending on the relative signs of , and , in contrast to the first-order calculation. To best interpret this behaviour we first note that for the Casimir geometry one has