# Geometrical dependence in Casimir-Polder repulsion: Anisotropically
polarizable atom and anisotropically polarizable annular dielectric

John Joseph Marchetta<EMAIL_ADDRESS>Department of Physics, Southern
Illinois University–Carbondale, Carbondale, Illinois 62901, USA  Prachi
Parashar<EMAIL_ADDRESS>John A. Logan College, Carterville,
Illinois 62918, USA  K. V. Shajesh<EMAIL_ADDRESS>Department of Physics,
Southern Illinois University–Carbondale, Carbondale, Illinois 62901, USA

###### Abstract

Casimir-Polder interaction energies between a point anisotropically
polarizable atom and an annular dielectric are shown to exhibit localized
repulsive long-range forces in specific configurations. We show that when the
atom is positioned at the center of the annular dielectric, it is
energetically favorable for the atom to align its polarizability with respect
to that of the dielectric. As the atom moves away from, but along the symmetry
axis of the annular dielectric, it encounters a point where the polarizable
atom experiences no torque and the energy is free of orientation dependence.
At this height, abruptly, the atom prefers to orient its polarizability
perpendicular to that of the dielectric. For certain configurations, it
encounters another torsion-free point a larger distance away, beyond which it
prefers to again point its polarizability with respect to that of the
dielectric. We find when the atom is close enough, and oriented such that the
energy is close to maximum, the atom could be repelled. For certain annular
polarizations, repulsion can happen below or above the torsion-free height.
Qualitative features differ when the atom is interacting with a ring, versus a
plate of infinite extent with a hole. In particular, the atom can prefer to
orient perpendicular to the polarizability of the plate at large distances, in
striking contrast to the expectation that it will orient parallel. To gain
insight of this discrepancy, we investigate an annular disc, which captures
the results of both geometries in limiting cases. These energies are too weak
for immediate applications, nevertheless, we elaborate an interesting
application on a prototype of a Casimir machine using these configurations.

##  I Introduction

The term Casimir effect has come to refer to the entire phenomena associated
with quantum fluctuations in electrodynamics. Figure 1 shows a schematic
diagram that identifies the various domains in interactions mediated by
quantum fluctuations. The van der Waals van der Waals (1873) and London
dispersion forces Eisenschitz and London (1930); London (1930); Hettema (2000)
govern these interactions in the non-retarded weak regime, while the Casimir-
Polder interaction energy Casimir and Polder (1948) involves interactions
between weak dielectric materials in the retarded regime. The energies in the
weak regime are often easier to compute. In contrast, the Casimir energy
between highly conducting materials in the retarded regime are typically hard
to compute because it involves identification and subtraction of divergences
in non-trivial ways to arrive at the expression for energy. The Casimir energy
is also very sought-after because it is a manifestation of zero point energy
associated to the quantum vacuum Casimir (1948); Milton (2011). The limiting
forms in Fig. 1 are generalized by the Lifshitz formula Lifshitz (1956), and
were further generalized and studied by Dzyaloshinskii, Lifshitz, and
Pitaevskii (DLP) Dzyaloshinskii  _et al._ (1961).

The Casimir effect most often leads to attractive forces between objects, and
the plausibility of associated repulsive forces has always captivated
attention. In the non-retarded van der Waals regime of Fig. 1, repulsion has
been predicted in three-body interactions at least since the work of Axilrod
and Teller Axilrod and Teller (1943) and Muto Muto (1943). In the Casimir-
Polder regime, repulsion between anisotropically polarizable atoms was
predicted in the works of Craig and Power Craig and Power (1969a, b) and has
been extensively explored since then; see Ref. Babb (2005) and references
therein. However, it was not until a decade ago when it was realized that
similar repulsive behavior was plausible in the Casimir regime of Fig. 1. This
came by in Ref. Levin  _et al._ (2010), where it was argued on physical
grounds that the interaction energy between an elongated needle-shaped-
conductor and a perfectly conducting metal sheet with a circular aperture
could have a local minima. They showed that the Casimir force could become
repulsive when the needle got sufficiently close to the aperture. This also
meant that the force between a charge placed in front a perfectly conducting
object could be repulsive Levin and Johnson (2011), and similar configurations
were further explored in Ref. McCauley  _et al._ (2011). It was realized there
that the anisotropy in the geometry of a highly conducting object corresponds
to an effective anisotropic permittivity and the interplay of these
anisotropic permittivities leads to non-monotonic interaction energies that
show repulsion. These remarkable predictions were based on symmetry arguments
and were confirmed using numerical calculations.

Here we list some of the attempts made towards understanding the result in
Ref. Levin  _et al._ (2010) analytically. In Ref.Eberlein and Zietal (2011),
it was shown that even in the non-retarded van der Waals regime, similar
configurations lead to repulsion. They used the method of inversion that was
successfully used in the heyday of electrostatics. In Ref. Abrantes  _et al._
(2018), the method of inversion was again used to show that in the non-
retarded van der Waals regime an anisotropically polarizable atom placed along
the symmetry axis of a toroid could experience repulsion. In Refs. Shajesh and
Schaden (2012) and Milton  _et al._ (2012a), it was shown that in the retarded
Casimir-Polder regime of Fig. 1, repulsion is possible in configurations with
anisotropically polarizable atoms and anisotropic dielectric materials. An
analytic formula for the Casimir-Polder energy between dielectric bodies was
derived in Refs. Shajesh and Schaden (2012) and Milton  _et al._ (2012a). In
particular, a closed form expression for the force between an anisotropically
polarizable atom and a dielectric plate with a circular aperture was shown to
demonstrate repulsion. In spite of the above successes in the weak scenario,
and having numerically found configurations exhibiting repulsion in the
Casimir regime of Fig. 1, analytic derivation of a closed-form expression for
the interaction energy between a polarizable atom and a highly conducting
plate with an aperture still remains an open problem. Initiative towards such
a derivation has been presented in Refs. Milton  _et al._ (2011) and Milton
_et al._ (2012b). Another proposal was to exploit the axial symmetry of the
configuration Shajesh  _et al._ (2017). However, a closed-form analytic result
remains elusive.

Figure 1: Diagram showing terminology associated with interactions mediated by
quantum fluctuations in electrodynamics. Here $\chi$ is the dielectric
susceptibility of a material characterized by plasma frequency $\omega_{p}$,
$\omega_{0}$ is a characteristic frequency of the material, $c$ is speed of
light in vacuum, and $r$ is the distance between two objects.

In this paper we investigate the interaction energy between two neutral
anisotropically polarizable objects mediated by quantum fluctuations. We
ignore magnetization effects here. The outline of the paper and summary of
results are as follows: In Sec. II, we derive the interaction energy between a
point permanent dipole and an electrically polarized ring. We give a summary
of the formalism in Sec. III, and present a few more configurations related to
annular plate in Sec. IV. We take up the study of a point anisotropically
polarizable atom and an anisotropically polarizable dielectric ring using
methods of Ref. Shajesh and Schaden (2012) in Sec. V. We show that for
specific orientations of the polarizability, the atom experiences a repulsive
force when the atom is very close to the ring. We analyze the orientation
dependence and distance dependence of the interaction energies. We also
observe that for specific orientations the atom experiences a second repulsive
region along the symmetry axis, which is a new finding here. In Sec. VI, we
derive the interaction energy between an atom and a dielectric annular disc.
For large outer radius, this energy approaches the interaction energy between
an atom and a plate with circular aperture. When the outer radius and inner
radius of the annular disc are close to each other, the energy approaches that
of a ring. In Sec. VII, we highlight and summarize our results. We point out
that there exists two torsion free points on each side of the annular disc,
where the interaction energy is orientation independent. The position of these
points vary with geometric changes to the disk, and differently according to
the polarizability; determining large distance orientation dependence. In the
ring limit of the annular disc, new repulsion emerges. A brief summary of our
results here has been separately presented in Ref. Marchetta  _et al._ (2020),
and the exposition here serves as supplementary material for the article. In
Sec. VIII, we describe an application of these results in the construction of
a Casimir machine. We give concluding remarks in Sec. IX.

For completeness, we list two other classes of repulsive behaviors that are
possible between neutral polarizable objects. One of these was mentioned in
Ref. Dzyaloshinskii  _et al._ (1961) in which repulsion is possible whenever
the dielectric strength of the intervening medium is stronger relative to the
surrounding media. The second is associated to repulsion arising from an
interplay of electric and magnetic properties of the neutral objects Feinberg
and Sucher (1968). In the current level of our understanding, apparently,
these repulsions have different origins from those discussed in this paper.
The repulsion studied in this article is geometric in origin due to the
presumable direct relationship between the effective anisotropies in the
dielectric susceptibility and the respective anisotropies in the geometric
shape of the conductor. The repulsive nature of self interaction of a single
conducting sphere Boyer (1968) could also probably be related to this
geometric origin, however it is not an interaction energy between two objects
as it is for cases we are considering here.

##  II Electrostatic analog: Point electric dipole above an electrically
polarized ring

Let us begin with an example involving neutral objects having permanent
electric polarizations. It will be a prelude for the following discussion
involving only neutral induced dipoles.

A point (permanent) electric dipole ${\bf p}$ at position ${\bf r}_{0}$ is
suitably described by the charge density

$\rho_{1}({\bf x})=-{\bf p}\cdot{\bm{\nabla}}\delta^{(3)}({\bf x}-{\bf
r}_{0}),$ (1)

and a (permanent) electrically polarized ring of radius $a$ with dipole per
unit length or polarization ${\bm{\lambda}}$ is described by the charge
density

$\rho_{2}({\bf
x}^{\prime})=-{\bm{\lambda}}\cdot{\bm{\nabla}}^{\prime}\delta(z^{\prime}-z_{a})\delta(\rho^{\prime}-a),$
(2)

such that the plane of the ring is perpendicular to the $z$ axis at
$z^{\prime}=z_{a}$ with center of the ring on the $z$ axis, see inset in Fig.
2. The electrostatic interaction energy between the point dipole and the ring,
in SI units, is given by

$E_{12}=\frac{1}{4\pi\varepsilon_{0}}\int d^{3}x\int
d^{3}x^{\prime}\frac{\rho_{1}({\bf x})\rho_{2}({\bf x}^{\prime})}{|{\bf
x}-{\bf x}^{\prime}|}.$ (3)

Five of the six integrals are immediately completed using the property of
$\delta$-functions appearing in the charge densities in Eqs. (1) and (2). This
leads to the interaction energy between the point dipole and the ring

$E_{12}=\frac{1}{4\pi\varepsilon_{0}}\int_{0}^{2\pi}ad\phi^{\prime}\frac{\left[({\bf
p}\cdot{\bm{\lambda}})-3({\bf p}\cdot\hat{\bf r})(\hat{\bf
r}\cdot{\bm{\lambda}})\right]}{r^{3}},$ (4)

where we have set

${\bf r}={\bf x}-{\bf x}^{\prime}.$ (5)

The magnitude of ${\bf r}$ is $r$ and the unit vector $\hat{\bf r}={\bf r}/r$.
Here ${\bf r}$ is the position of the electric dipole relative to the position
of a line element $ad\phi^{\prime}$ on the ring that can be explicitly
expressed using

$\displaystyle{\bf x}$ $\displaystyle=$ $\displaystyle\rho\cos\phi\,\hat{\bf
x}+\rho\sin\phi\,\hat{\bf y}+(z_{a}+h)\,\hat{\bf z},$ (6a) $\displaystyle{\bf
x}^{\prime}$ $\displaystyle=$ $\displaystyle a\cos\phi^{\prime}\,\hat{\bf
x}+a\sin\phi^{\prime}\,\hat{\bf y}+z_{a}\,\hat{\bf z},$ (6b)

${\bf r}_{0}={\bf x}$, such that

$r^{2}=a^{2}+h^{2}+\rho^{2}-2a\rho\cos(\phi-\phi^{\prime}).$ (7)

For the particular case when the ring is polarized in the tangential direction
$\hat{\bm{\phi}}^{\prime}$,

${\bm{\lambda}}=\lambda\hat{\bm{\phi}}^{\prime},$ (8)

and the position of the point dipole is confined on the symmetry axis of the
ring, $\rho=0$, the interaction energy is

$E_{12}=0$ (9)

irrespective of the direction of its dipole moment ${\bf p}$. This is so
because $(\hat{\bf r}\cdot{\bm{\lambda}})=\lambda(\hat{\bf
r}\cdot\hat{\bm{\phi}}^{\prime})=0$ and the integration

$\int_{0}^{2\pi}d\phi^{\prime}\,({\bf p}\cdot{\bm{\lambda}})=0$ (10)

in this case, which involves integration of a cosine function over its
complete period.

For the case when the polarization of the ring is aligned to its symmetry
axis, and the electric dipole ${\bf p}$ is positioned on the axis and the
direction of ${\bf p}$ makes an angle $\theta$ with respect to the symmetry
axis,

${\bm{\lambda}}=\lambda\,\hat{\bf z},\qquad\hat{\bm{\lambda}}\cdot\hat{\bf
p}=\cos\theta,\qquad\rho=0.$ (11)

The interaction energy in this case is

$E_{12}=\frac{{\bf
p}\cdot{\bm{\lambda}}}{a^{2}}\frac{2\pi}{4\pi\varepsilon_{0}}\frac{a^{3}(a^{2}-2h^{2})}{\left(a^{2}+h^{2}\right)^{\frac{5}{2}}}.$
(12)

Figure 2:  Electrostatic interaction energy between a point electric dipole
${\bf p}$ and a polarized ring of polarization ${\bm{\lambda}}$ as given by
Eq. (12). Here $E_{12}$ is plotted with respect to $h$ for
$\theta=0^{\circ},30^{\circ},60^{\circ},75^{\circ}$. The energy at $\theta=0$
and $h=0$ is $p\lambda 2\pi/a^{2}4\pi\varepsilon_{0}$.

The force in the $z$-direction, a manifestation of change in energy with
respect to $h$, is

${\bf F}\cdot\hat{\bf z}=-\frac{\partial E_{12}}{\partial h}=\frac{{\bf
p}\cdot{\bm{\lambda}}}{a^{3}}\frac{2\pi}{4\pi\varepsilon_{0}}\frac{3a^{4}h(3a^{2}-2h^{2})}{\left(a^{2}+h^{2}\right)^{\frac{7}{2}}}$
(13)

and the torque, a manifestation of change in energy with respect $\theta$, is

${\bm{\tau}}=-\frac{\partial E_{12}}{\partial\theta}=\frac{{\bf
p}\times{\bm{\lambda}}}{a^{2}}\frac{2\pi}{4\pi\varepsilon_{0}}\frac{a^{3}(a^{2}-2h^{2})}{\left(a^{2}+h^{2}\right)^{\frac{5}{2}}}.$
(14)

The interaction energy of Eq. (12) is plotted as a function of $h$ for
different values of orientation angle $\theta$ in Fig. 2. The polarizations of
the dipole and the ring prefer to align anti-parallelly for $h\ll a$ and
parallelly for $h\gg a$. The energy is zero and torsion-free when
$h=a/\sqrt{2}$, represented by the point where all the curves intersect in
Fig. 2.

##  III Formalism

.

We now turn our attention to the interaction between neutral polarizable
objects. As mentioned in Introduction, the interaction energy between
polarizable materials is governed by the van der Waals and London interaction
van der Waals (1873); Eisenschitz and London (1930); London (1930), which
generalizes to the Casimir and the Lifshitz interaction Casimir (1948);
Lifshitz (1956); Dzyaloshinskii  _et al._ (1961) when retardation effects are
taken into account. In the following, we illustrate the formalism. This
section is pedagogical and aimed towards listing various approximations in the
discussion.

The Casimir or Lifshitz interaction energy between two dielectric materials
can be expressed as Shajesh and Schaden (2011)

$E_{12}=\frac{\hbar
c}{2}\int_{-\infty}^{\infty}\frac{d\zeta}{2\pi}\,\text{Tr}\ln\Big{[}{\bf
1}-{\bf\Gamma}_{0}\cdot{\bf T}_{1}\cdot{\bf\Gamma}_{0}\cdot{\bf T}_{2}\Big{]}$
(15)

in the multiple scattering formalism, where $i\zeta=\omega/c$ is the imaginary
frequency and $\omega$ is the frequency. The free Green’s dyadic
${\bf\Gamma}_{0}$ in Eq. (15) satisfies the dyadic differential equation

$-\Big{[}\frac{1}{\zeta^{2}}{\bm{\nabla}}\times({\bm{\nabla}}\times{\bf
1})+{\bf 1}\Big{]}\cdot{\bm{\Gamma}}_{0}({\bf r};i\zeta)={\bf
1}\,\delta^{(3)}({\bf r})$ (16)

with solution

${\bf\Gamma}_{0}({\bf
r};i\zeta)=\frac{e^{-|\zeta|r}}{4\pi\,r^{3}}\Big{[}-u(|\zeta|r)\,{\bf
1}+v(|\zeta|r)\,\hat{\bf r}\hat{\bf r}\Big{]},$ (17)

where

$\displaystyle u(x)$ $\displaystyle=$ $\displaystyle 1+x+x^{2},$ (18a)
$\displaystyle v(x)$ $\displaystyle=$ $\displaystyle 3+3x+x^{2}.$ (18b)

The electric susceptibilities of the dielectric materials, of electric
permittivity ${\bm{\varepsilon}}_{i}$, is

${\bm{\chi}}_{i}({\bf x};i\zeta)=\frac{{\bm{\varepsilon}}_{i}({\bf
x};i\zeta)}{\varepsilon_{0}}-{\bf 1},$ (19)

where $i=1,2$, denotes respective material. In terms of electric
susceptibilities the transition matrices ${\bf T}_{i}$’s in Eq. (15) are
symbolically defined by the relation

${\bf T}_{i}={\bm{\chi}}_{i}\cdot\big{[}{\bf
1}-{\bm{\Gamma}}_{0}\cdot{\bm{\chi}}_{i}\big{]}^{-1}.$ (20)

The transition matrices are themselves local quantities, but for mathematical
purposes are represented as a kernel,

${\bf T}_{i}({\bf x},{\bf x}^{\prime};i\zeta)={\bf T}_{i}({\bf
x};i\zeta)\,\delta^{(3)}({\bf x}-{\bf x}^{\prime}).$ (21)

The transition matrices are determined by solving the Green’s dyadic equation

$-\Big{[}\frac{1}{\zeta^{2}}{\bm{\nabla}}\times({\bm{\nabla}}\times{\bf
1})+{\bf 1}+{\bm{\chi}}({\bf x};i\zeta)\Big{]}\cdot{\bm{\Gamma}}_{i}({\bf
x},{\bf x}^{\prime};i\zeta)={\bf 1}\,\delta^{(3)}({\bf x}-{\bf x}^{\prime}),$
(22)

where ${\bm{\Gamma}}_{i}({\bf x},{\bf x}^{\prime};i\zeta)$ are the Green’s
dyadic for the individual materials

${\bm{\Gamma}}_{i}={\bm{\Gamma}}_{0}-{\bm{\Gamma}}_{0}\cdot{\bf
T}_{i}\cdot{\bm{\Gamma}}_{0}.$ (23)

###  III.1 Point atom

Neglecting quadruple and higher moments, the scattering matrix ${\bf
T}_{\text{atom}}$ for an atom with atomic dipole polarizability
${\bm{\alpha}}(i\zeta)$ will be modeled as

${\bf T}_{\text{atom}}({\bf
x};i\zeta)=4\pi{\bm{\alpha}}(i\zeta)\,\delta^{(3)}({\bf x}-{\bf x}_{0}),$ (24)

where ${\bf x}_{0}$ specifies the position of the atom. The polarizabilities
are frequency dependent. The $\delta$-functions in Eq. (24) permit a trivial
evaluation of half of the spatial integrals in the trace of Eq. (15). The
frequency dependence of the atomic polarizability ${\bm{\alpha}}(i\zeta)$ is
in general very complicated and depends on the energy levels of the individual
atom. For example, if one assumes a two-level atom with a spherically
symmetric ground state and a spherically asymmetric excited state, then a
simple model for the frequency dependence of the atomic polarizability is

${\bm{\alpha}}(i\zeta)={\bm{\alpha}}(0)\frac{\omega_{0}^{2}}{\omega_{0}^{2}+\zeta^{2}c^{2}},$
(25)

where $\omega_{0}$ is the excitation energy of the (two-level) atom and
${\bm{\alpha}}(0)$ is its static polarizability and a measure of the spherical
asymmetry of the excited state. In terms of the radius $a_{0}$ of this simple
(two-level) atom, the atomic polarizability could be further modeled as

${\bm{\alpha}}(0)=\tilde{\bm{\alpha}}_{0}\,a_{0}^{3}.$ (26)

In this manner, one captures the content of the spherical asymmetry in the
excited state in the dimensionless parameter $\tilde{\bm{\alpha}}_{0}$.
Formally, we are ignoring the detailed structure of the transition matrix for
an atom by replacing Eq. (20) in place with Eq. (24), and is valid for
separation distances satisfying

$|\tilde{\bm{\alpha}}_{0}|\,a_{0}\ll r,$ (27)

because the propagators inside Eq. (20) rarely span distances greater than the
atomic size. This approximation is valid for dilute polarizability,
$|\tilde{\bm{\alpha}}_{0}|\ll 1$, and atomic size small compared to separation
distance. In this atomic approximation one needs to retain only the leading
term of the logarithm after expansion of Eq. (15),

$\displaystyle E_{\text{atom}}^{\text{W}}({\bm{\alpha}},{\bm{\chi}};{\bf
x}_{0})=-\hbar c\int_{-\infty}^{\infty}d\zeta\int d^{3}x\,\hskip 71.13188pt$
$\displaystyle\,\text{tr}\,{\bm{\alpha}}(i\zeta)\cdot{\bm{\Gamma}}({\bf
x}_{0}-{\bf x};i\zeta)\cdot{\bf T}({\bf x};i\zeta)\cdot{\bm{\Gamma}}_{0}({\bf
x}-{\bf x}_{0};i\zeta),\hskip 14.22636pt$ (28)

where the trace, tr, is over the dyadic index, and ${\bf T}$ is the transition
matrix of the dielectric material that the atom is interacting with.
Substituting the free Green’s dyadic from Eq. (17) into Eq. (28) we have

$\displaystyle E_{\text{atom}}^{\text{W}}({\bm{\alpha}},{\bm{\chi}};{\bf
x}_{0})=-\frac{\hbar c}{16\pi^{2}}\int_{-\infty}^{\infty}d\zeta\int
d^{3}x\,\frac{e^{-2|\zeta|r}}{r^{6}}\hskip 28.45274pt$ $\displaystyle\hskip
28.45274pt\times\Big{[}u^{2}(\zeta
r)\,\text{tr}\\{{\bm{\alpha}}(i\zeta)\cdot{\bf T}({\bf x};i\zeta)\\}$
$\displaystyle\hskip 39.83385pt-2u(\zeta r)v(\zeta r)\,\\{\hat{\bf
r}\cdot{\bm{\alpha}}(i\zeta)\cdot{\bf T}({\bf x};i\zeta)\cdot\hat{\bf r}\\}$
$\displaystyle\hskip 39.83385pt+v^{2}(\zeta r)\,\\{\hat{\bf
r}\cdot{\bm{\alpha}}(i\zeta)\cdot\hat{\bf r}\\}\\{\hat{\bf r}\cdot{\bf T}({\bf
x};i\zeta)\cdot\hat{\bf r}\\}\Big{]},$ (29)

where ${\bf r}={\bf x}-{\bf x}_{0}$, and $r=|{\bf r}|$. We used the property
that atomic polarizability ${\bm{\alpha}}$ and the transition matrix ${\bf T}$
are symmetric tensors that let us combine the two cross terms into a single
one.

###  III.2 van der Waals-London approximation

The non-retarded (van der Waals-London) regime,

$|{\bm{\alpha}}(0)|^{1/3}<r\ll\frac{c}{\omega_{0}},$ (30)

is a short-range approximation, where short is in relation to the
characteristic length associated with resonant frequency. In the regime of Eq.
(30), the frequency dependence in Eq. (17) may be neglected and the free
dyadic in Eq. (28) is approximated by the static dipole-dipole interaction
${\bm{\Gamma}}_{0}({\bm{r}};0)$ to yield Craig and Power (1969a, b), using Eq.
(28),

$\displaystyle E_{\text{atom}}^{\text{Lon}}({\bm{\alpha}},{\bm{\chi}};{\bf
x}_{0})=\hbar c\int_{-\infty}^{\infty}d\zeta\int d^{3}x\hskip 71.13188pt$
$\displaystyle\hskip
11.38109pt\,\text{tr}\,{\bm{\alpha}}(i\zeta)\cdot{\bm{\Gamma}}({\bf
x}_{0}-{\bf x};0)\cdot{\bf T}({\bf x};i\zeta)\cdot{\bm{\Gamma}}({\bf x}-{\bf
x}_{0};0).$ (31)

In this the way, Eq. (29) simplifies to

$\displaystyle E_{\text{atom}}^{\text{Lon}}({\bm{\alpha}},{\bm{\chi}};{\bf
x}_{0})=-\frac{\hbar c}{16\pi^{2}}\int_{-\infty}^{\infty}d\zeta\int
d^{3}x\,\frac{1}{r^{6}}$ $\displaystyle\hskip
28.45274pt\times\Big{[}\,\text{tr}\\{{\bm{\alpha}}(i\zeta)\cdot{\bf T}({\bf
x};i\zeta)\\}-6\,\\{\hat{\bf r}\cdot{\bm{\alpha}}(i\zeta)\cdot{\bf T}({\bf
x};i\zeta)\cdot\hat{\bf r}\\}$ $\displaystyle\hskip 39.83385pt+9\,\\{\hat{\bf
r}\cdot{\bm{\alpha}}(i\zeta)\cdot\hat{\bf r}\\}\\{\hat{\bf r}\cdot{\bf T}({\bf
x};i\zeta)\cdot\hat{\bf r}\\}\Big{]},$ (32)

which uses $u(0)=1$ and $v(0)=3$. For two atoms with isotropic
polarizabilities, this reproduces London’s expression for the van der Waals
interaction

$E_{\text{atom-atom}}^{\text{Lon}}(\alpha_{1}{\bf 1},\alpha_{2}{\bf
1};r)=-\frac{3\hbar
c}{\pi\,r^{6}}\int_{0}^{\infty}d\zeta\,\alpha_{1}(i\zeta)\alpha_{2}(i\zeta),$
(33)

which is inversely proportional to the sixth power in the separation $r$.

###  III.3 Casimir-Polder approximation

The retarded (Casimir-Polder) regime,

$|{\bm{\alpha}}(0)|^{1/3}<\frac{c}{\omega_{0}}\ll r,$ (34)

is the corresponding long-range approximation when only fluctuations of very
large time period (small frequency) contribute to resonance. In this
approximation, the exponential dependence on the separation distance in the
free propagator of Eq. (17) implies that the frequency dependence of the
polarizability ${\bm{\alpha}_{i}}(i\zeta)$ in Eq. (25) is negligible. In the
Casimir-Polder regime, the polarizabilities can be approximated by their
static values in Eq. (28) to yield Craig and Power (1969a, b),

$\displaystyle E_{\text{atom}}^{\text{CP}}({\bm{\alpha}},{\bm{\chi}};{\bf
x}_{0})=-\hbar c\int_{-\infty}^{\infty}d\zeta\int d^{3}x\,\hskip 56.9055pt$
$\displaystyle\hskip
14.22636pt\times\,\text{tr}\,{\bm{\alpha}}(0)\cdot{\bm{\Gamma}}({\bf
x}_{0}-{\bf x};i\zeta)\cdot{\bf T}({\bf x};0)\cdot{\bm{\Gamma}}({\bf x}-{\bf
x}_{0};i\zeta).\hskip 14.22636pt$ (35)

The $\zeta$-integration in Eq. (29) can be performed in this approximation to
obtain

$\displaystyle E_{\text{atom}}^{\text{CP}}({\bm{\alpha}},{\bm{\chi}};{\bf
x}_{0})=-\frac{\hbar c}{32\pi^{2}}\int d^{3}x\,\frac{1}{r^{7}}$
$\displaystyle\hskip
14.22636pt\times\Big{[}13\,\text{tr}\\{{\bm{\alpha}}(0)\cdot{\bf T}({\bf
x};0)\\}-56\,\\{\hat{\bf r}\cdot{\bm{\alpha}}(0)\cdot{\bf T}({\bf
x};0)\cdot\hat{\bf r}\\}$ $\displaystyle\hskip 25.60747pt+63\,\\{\hat{\bf
r}\cdot{\bm{\alpha}}(0)\cdot\hat{\bf r}\\}\\{\hat{\bf r}\cdot{\bf T}({\bf
x};0)\cdot\hat{\bf r}\\}\Big{]}.$ (36)

For atoms with isotropic polarizabilities ${\bm{\alpha}}_{1}=\alpha_{1}{\bf
1}$ and ${\bm{\alpha}}_{2}=\alpha_{2}{\bf 1}$, Eq. (36) reproduces the
Casimir-Polder interaction Casimir and Polder (1948)

$E_{12}^{\text{CP}}(\alpha_{1}{\bf 1},\alpha_{2}{\bf 1};r)=-\frac{23\hbar
c}{4\pi}\frac{\alpha_{1}(0)\alpha_{2}(0)}{r^{7}},$ (37)

which is inversely proportional to the seventh power in the separation $r$.

###  III.4 Dilute dielectric approximation

For a dilute dielectric material, we have

${\bm{\chi}}\ll{\bf 1}.$ (38)

Thus, after keeping the leading term in the expansion of Eq. (20), the
scattering matrix for the dilute material becomes

${\bf T}_{i}({\bf x},{\bf x}^{\prime};i\zeta)\sim{\bm{\chi}}_{i}({\bf
x};i\zeta)\,\delta^{(3)}({\bf x}-{\bf x}^{\prime}).$ (39)

The dilute dielectric approximation of Eq. (28) is

$\displaystyle E_{\text{atom-dd}}^{\text{W}}=-\hbar
c\int_{-\infty}^{\infty}d\zeta\int d^{3}x$ $\displaystyle\hskip
8.53581pt\,\text{tr}\,{\bm{\alpha}}(i\zeta)\cdot{\bm{\Gamma}}_{0}({\bf
x}_{0}-{\bf x};i\zeta)\cdot{\bm{\chi}}({\bf
x};i\zeta)\cdot{\bm{\Gamma}}_{0}({\bf x}-{\bf x}_{0};i\zeta).\hskip
19.91692pt$ (40)

The London interaction energies in Eqs. (31) and (32), and the Casimir-Polder
energies in Eqs. (35) and (36), in the approximation of Eq. (38) are
immediately obtained by the replacement: ${\bf T}\to{\bm{\chi}}$. For example,
the Casimir-Polder interaction energy between an atom and a dilute dielectric,
using Eq. (24) and Eq. (19), is given by

$\displaystyle E_{\text{atom-dd}}^{\text{CP}}({\bm{\alpha}},{\bm{\chi}};{\bf
x}_{0})=-\frac{\hbar c}{32\pi^{2}}\int d^{3}x\,\frac{1}{r^{7}}$
$\displaystyle\hskip
14.22636pt\times\Big{[}13\,\text{tr}\\{{\bm{\alpha}}(0)\cdot{\bm{\chi}}({\bf
x};0)\\}-56\,\\{\hat{\bf r}\cdot{\bm{\alpha}}(0)\cdot{\bm{\chi}}({\bf
x};0)\cdot\hat{\bf r}\\}$ $\displaystyle\hskip 25.60747pt+63\,\\{\hat{\bf
r}\cdot{\bm{\alpha}}(0)\cdot\hat{\bf r}\\}\\{\hat{\bf r}\cdot{\bm{\chi}}({\bf
x};0)\cdot\hat{\bf r}\\}\Big{]},$ (41)

which is a ready-to-use expression, because unlike the earlier approximations,
it does not require the solution for the transition matrix ${\bf T}$ obtained
by solving the Green’s dyadic equation in Eq. (22). The dilute-dielectric
approximation seems to always contain all the qualitative features of the
interaction energy, and thus serves well for a qualitative understanding of
complicated geometries.

##  IV Point atom above an infinite dielectric plate with a circular aperture

Figure 3:  The energy of Eq. (45) is plotted as a function of height $h$ for
an anisotropic atom above the center of a dielectric plate with an aperture of
radius $a$ as sketched in the insert. The curves correspond to different
orientation $\theta$ of the atomic polarizability.

As a demonstration of the use of Eq. (41), in Ref. Shajesh and Schaden (2012),
the energy expression of Eq. (41) was used to determine the interaction energy
of an atom with polarizability,

${\bm{\alpha}}=\alpha_{1}\,\hat{\bf e}_{1}\,\hat{\bf e}_{1},\quad\hat{\bf
e}_{1}\cdot\hat{\bf z}=\cos\theta,$ (42)

with a dielectric plate of infinite extent with a circular aperture of radius
$a$ with polarizability

${\bm{\chi}}={\bm{\lambda}}\,\theta(\sqrt{x^{2}+y^{2}}-a)\,\delta(z)$ (43)

when the atom is positioned on the symmetry axis of the plate.

###  IV.1 Partially isotropic plate

For the case when dilute dielectric plate has isotropic polarizability in the
plane of the plate we have

${\bm{\lambda}}=\lambda\big{[}\hat{\bf x}\hat{\bf x}+\hat{\bf y}\hat{\bf
y}\big{]}=\lambda\big{[}\hat{\bm{\rho}}\hat{\bm{\rho}}+\hat{\bm{\phi}}\hat{\bm{\phi}}\big{]}.$
(44)

The interaction energy has the closed form exact expression

$\displaystyle E$ $\displaystyle=$ $\displaystyle-\frac{\hbar
c\alpha_{1}\lambda}{64\pi}\frac{1}{5}\frac{1}{(a^{2}+h^{2})^{\frac{9}{2}}}\Big{[}(33a^{4}+106a^{2}h^{2}+38h^{4})$
(45) $\displaystyle\hskip 28.45274pt-(33a^{4}-74a^{2}h^{2}-2h^{4})\cos
2\theta\Big{]}.\hskip 17.07164pt$

Fig. 3 shows the dependence of Eq. (45) on the height $h$ for different
orientations $\theta$ of the anisotropic atom. The parameter space of height
$h$ and orientation angle $\theta$ that leads to repulsion has been shown as
shaded regions in Fig. 4. Repulsion ceases for orientations
$\theta>\cos^{-1}(17/89)/2\sim 39.49^{\circ}$. We refer to Ref. Shajesh and
Schaden (2012) for a detailed discussion. We point out that the expression for
interaction energy in Eq. (18) in Ref. Shajesh and Schaden (2012) has errors
in the coefficients there and should be replaced with the correct form in Eq.
(45). This error propagates into the analysis of repulsion there, however, all
the qualitative features of the result and associated conclusions remain
unchanged.

Figure 4: The shaded regions in the parameter space of height $h$ and
orientation angle $\theta$ above leads to repulsion between an anisotropic
atom at a height $h$ above the center of a dielectric plate with an aperture
of radius $a$ as sketched in the insert.

###  IV.2 Radially polarizable plate with aperture

When the dilute dielectric plate is polarizable in the radial directions, we
have

${\bm{\lambda}}=\lambda_{\rho}\hat{\bm{\rho}}\hat{\bm{\rho}}.$ (46)

The interaction energy has the form

$\displaystyle E$ $\displaystyle=$ $\displaystyle-\frac{\hbar
c\alpha_{1}\lambda_{\rho}}{64\pi}\frac{1}{(a^{2}+h^{2})^{\frac{9}{2}}}\Big{[}(4a^{4}+16a^{2}h^{2}+5h^{4})$
(47) $\displaystyle\hskip 28.45274pt-(4a^{4}-20a^{2}h^{2}-3h^{4})\cos
2\theta\Big{]}.\hskip 17.07164pt$

The qualitative features of energy plots is completely captured in the
corresponding energy plots for the partially isotropic case in Fig. 4. We
shall return to the expression for energy in Eq. (47) while discussing the
case of a circular disc with circular aperture as the outer radius of the disc
extends to infinity.

###  IV.3 Axially polarizable plate with aperture

When the dilute dielectric plate is polarizable in the direction of its
symmetry axis, we have

${\bm{\lambda}}=\lambda_{z}\hat{\bf z}\hat{\bf z}.$ (48)

The interaction energy has the form

$\displaystyle E$ $\displaystyle=$ $\displaystyle-\frac{\hbar
c\alpha_{1}\lambda_{z}}{64\pi}\frac{1}{5}\frac{1}{(a^{2}+h^{2})^{\frac{9}{2}}}\Big{[}(26a^{4}+17a^{2}h^{2}+26h^{4})$
(49) $\displaystyle\hskip 28.45274pt+(26a^{4}-73a^{2}h^{2}+6h^{4})\cos
2\theta\Big{]}.\hskip 17.07164pt$

The energy has two orientation independent points for $h>0$ at $h\sim 0.61a$
and $h\sim 3.44a$. Again, we shall return to this while discussing a disc with
a circular aperture.

Figure 5:  The energy of Eq. (49) is plotted as a function of height $h$ for
an anisotropic atom above the center of a dielectric plate with an aperture of
radius $a$ with polarizability in the direction of symmetry axis of plate as
sketched in the insert. The curves correspond to different orientation
$\theta$ of the atomic polarizability. All the curves have two orientation
independent points for $h>0$ at $h\sim 0.61a$ and $h\sim 3.44a$, with the
second orientation independent point visible only under sufficient zoom level.

##  V Point atom above a dielectric ring

The Casimir-Polder interaction energy between an atom and a dielectric, Eq.
(24) in Eq. (41), is given by the expression

$\displaystyle E=-\frac{\hbar c}{32\pi^{2}}\int
d^{3}x\frac{1}{r^{7}}\Big{[}13\,\text{tr}({\bm{\alpha}}\cdot{\bm{\chi}})-56\,(\hat{\bf
r}\cdot{\bm{\alpha}}\cdot{\bm{\chi}}\cdot\hat{\bf r})$ $\displaystyle\hskip
96.73936pt+63\,(\hat{\bf r}\cdot{\bm{\alpha}}\cdot\hat{\bf r})(\hat{\bf
r}\cdot{\bm{\chi}}\cdot\hat{\bf r})\Big{]}$ (50)

where

${\bf r}={\bf x}-{\bf x}_{0},$ (51)

${\bf x}_{0}$ being the position of the atom and ${\bf x}$ being the integral
variable scanning the infinitesimal elements of the dielectric. The dielectric
function for the atom is described by the atomic polarizability
${\bm{\alpha}}$ and the electric susceptibility for the interacting material
is ${\bm{\chi}}$. We will confine our discussion to the retarded Casimir-
Polder regime of Sec. III.3, where only the static zero frequency modes in
polarizability contribute.

Let the ring be assumed to be on the $xy$-plane with its center at the origin.
Thus, the electric susceptibility of the ring is

${\bm{\chi}}={\bm{\sigma}}\delta(z-0)\delta(\rho-a),$ (52)

where $a$ is the radius of the ring and ${\bm{\sigma}}$ is the polarizability
of the ring. Let the atom with polarizability ${\bm{\alpha}}$ be positioned on
the symmetry axis of the ring at a height $h$ above the center of the ring.
Thus, the electric susceptibility of the atom is

${\bm{\chi}}={\bm{\alpha}}\delta^{(3)}({\bf x}-{\bf x}_{0}),$ (53)

where

${\bf x}_{0}=h\,\hat{\bf z}.$ (54)

See Fig. 6 for an illustrative diagram.

Figure 6: Point atom of polarizability ${\bm{\alpha}}$ above a dielectric ring
of polarizability ${\bm{\sigma}}$. The atom is on the symmetry axis of the
ring.

Since the atom is placed on the symmetry axis of the ring, we have the
simplification

${\bf r}=a\hat{\bm{\rho}}-h\hat{\bf z}$ (55)

and

$r=\sqrt{a^{2}+h^{2}},\qquad\hat{\bf r}=\frac{a\hat{\bm{\rho}}-h\hat{\bf
z}}{\sqrt{a^{2}+h^{2}}}.$ (56)

Thus, integrating over $\rho$ and $z$, the expression for the interaction
energy in Eq. (50) takes the simplified form

$\displaystyle E$ $\displaystyle=$ $\displaystyle-\frac{\hbar
c}{32\pi^{2}}\frac{a}{(a^{2}+h^{2})^{\frac{7}{2}}}\int_{0}^{2\pi}d\phi\Big{[}13\,\text{tr}({\bm{\alpha}}\cdot{\bm{\sigma}})$
(57) $\displaystyle-56\,(\hat{\bf
r}\cdot{\bm{\alpha}}\cdot{\bm{\sigma}}\cdot\hat{\bf r})+63\,(\hat{\bf
r}\cdot{\bm{\alpha}}\cdot\hat{\bf r})(\hat{\bf
r}\cdot{\bm{\sigma}}\cdot\hat{\bf r})\Big{]}.\hskip 28.45274pt$

We emphasize that the above expression assumes the following, (mostly for
simplification in the ensuing analysis,) in addition to the approximations
clarified in Section III.

  1. 1.

We assume that the position of the point atom is exactly on the symmetry axis
of the ring, while the orientations of the principal axes of polarizability
are kept arbitrary. This puts a severe limitation if we were interested in the
stability analysis of the atom.

  2. 2.

We assume that the polarizability tensor of the ring is diagonal in the
eigenbasis used to the characterize the geometry of the ring. These are chosen
to be $\hat{\bf z}$, $\hat{\bm{\rho}}$, and $\hat{\bm{\phi}}$, with the
direction of $\hat{\bf z}$ chosen along the symmetry axis of the ring. That
is, we will assume that the polarizability tensor of the ring has the form

${\bm{\sigma}}=\sigma_{z}\,\hat{\bf z}\hat{\bf
z}+\sigma_{\rho}\,\hat{\bm{\rho}}\hat{\bm{\rho}}+\sigma_{\phi}\,\hat{\bm{\phi}}\hat{\bm{\phi}}.$
(58)

We allow the polarizability tensor of the atom to be completely arbitrary,

${\bm{\alpha}}=\alpha_{1}\hat{\bf e}_{1}\hat{\bf e}_{1}+\alpha_{2}\hat{\bf
e}_{2}\hat{\bf e}_{2}+\alpha_{3}\hat{\bf e}_{3}\hat{\bf e}_{3}.$ (59)

Even with the above assumptions, the analysis involves sufficient number of
parameters required to specify the anisotropies of the polarizations relative
to the vector describing the distance between the materials. The principal
axes for the ring are suitably chosen to be

$\displaystyle\hat{\bm{\rho}}$ $\displaystyle=$
$\displaystyle\cos\phi\,\hat{\bf i}+\sin\phi\,\hat{\bf j},$ (60a)
$\displaystyle\hat{\bm{\phi}}$ $\displaystyle=$
$\displaystyle-\sin\phi\,\hat{\bf i}+\cos\phi\,\hat{\bf j},$ (60b)
$\displaystyle\hat{\bf z}$ $\displaystyle=$ $\displaystyle\hat{\bf k},$ (60c)

which are unit vectors associated to the cylindrical polar coordinates. The
principal axes for the atom are kept arbitrary to allow complete generality by
choosing

$\displaystyle\hat{\bf e}_{1}$ $\displaystyle=$ $\displaystyle\hat{\bf
r}_{s},$ (61a) $\displaystyle\hat{\bf e}_{2}$ $\displaystyle=$
$\displaystyle\cos\beta\,\hat{\bm{\phi}}_{s}+\sin\beta\,\hat{\bm{\theta}}_{s},$
(61b) $\displaystyle\hat{\bf e}_{3}$ $\displaystyle=$
$\displaystyle-\sin\beta\,\hat{\bm{\phi}}_{s}+\cos\beta\,\hat{\bm{\theta}}_{s},$
(61c)

where $\beta$ is an angle of rotation about the axes $\hat{\bf e}_{1}$. Unit
vectors $\hat{\bf r}_{s}$, $\hat{\bm{\theta}}_{s}$, and $\hat{\bm{\phi}}_{s}$
are the unit vectors associated to the spherical polar coordinates,

$\displaystyle\hat{\bf r}_{s}$ $\displaystyle=$
$\displaystyle\sin\theta\cos\phi_{s}\,\hat{\bf
i}+\sin\theta\sin\phi_{s}\,\hat{\bf j}+\cos\theta\,\hat{\bf k},$ (62a)
$\displaystyle\hat{\bm{\theta}}_{s}$ $\displaystyle=$
$\displaystyle\cos\theta\cos\phi_{s}\,\hat{\bf
i}+\cos\theta\sin\phi_{s}\,\hat{\bf j}-\sin\theta\,\hat{\bf k},$ (62b)
$\displaystyle\hat{\bm{\phi}}_{s}$ $\displaystyle=$
$\displaystyle-\sin\phi_{s}\,\hat{\bf i}+\cos\phi_{s}\,\hat{\bf j}.$ (62c)

Since the axes $\hat{\bf e}_{1}$ need not be normal to the $x$-$y$ plane, we
choose the spherical coordinate $\phi_{s}$ to be different from the
cylindrical coordinate $\phi$. The configuration of an anisotropic atom above
an anisotropic ring in terms of the above parameters involves the integration
over the angle $\phi$, which renders the interaction energy to be dependent on
the azimuth angle $\beta$ and independent of $\phi_{s}$ and $\phi$. This
characteristic allows the choice $\phi_{s}=0$, however, we shall refrain and
allow the algebra to bring out this feature explicitly.

###  V.1 Tangentially polarizable ring

To illustrate the anisotropic features of the interaction in Eq. (50) we begin
by considering a special case. It will turn out that this case does not permit
repulsion. Nevertheless, the associated simplification in the configuration
brings out the role of anisotropy in the interaction.

Let the polarizability of the ring be purely in the tangential direction, that
is,

${\bm{\sigma}}=\sigma_{\phi}\,\hat{\bm{\phi}}\hat{\bm{\phi}},$ (63)

where $\hat{\bm{\phi}}$ is a unit vector tangent to the ring. Then, because
this vector is always perpendicular to the relative position vector ${\bf r}$,

$\hat{\bm{\phi}}\cdot\hat{\bf r}=0,$ (64)

using Eq. (56) we conclude that two of the three modes of interaction in Eq.
(57) do not contribute. Thus, we have

$\displaystyle E$ $\displaystyle=$ $\displaystyle-\frac{\hbar
c\sigma_{\phi}}{32\pi^{2}}\frac{13a}{(a^{2}+h^{2})^{\frac{7}{2}}}\int_{0}^{2\pi}d\phi\,(\hat{\bm{\phi}}\cdot{\bm{\alpha}}\cdot\hat{\bm{\phi}}).$
(65)

The construction $(\hat{\bm{\phi}}\cdot{\bm{\alpha}}\cdot\hat{\bm{\phi}})$
involves projection of the polarizability of the atom along one of the
directions $\hat{\bm{\phi}}$ associated with the ring, and thus is independent
of the height $h$. All the dependence in $h$ is contained in
$1/(a^{2}+h^{2})^{7/2}$. Hence, we conclude that the energy is a monotonic
function in its dependence in height $h$. This renders the component of the
force on the atom in the direction of $\hat{\bf z}$ to be always attractive,
because it is the negative derivative of energy with respect to $h$. That is,
a ring that is polarizable only in the tangential direction always attracts a
polarizable atom on the symmetry axis of the ring.

This conclusion leaves this particular case uninteresting in a discussion on
the repulsive Casimir force. However, as we mentioned earlier, we shall carry
out the discussion to a certain extent to highlight the role of anisotropy in
these interactions.

####  V.1.1 Case: ${\bf\sigma}_{\phi}$ and $\alpha_{1}$

Let the polarizability of the atom be uniaxial in an arbitrary direction
$\hat{\bf e}_{1}$ given by

${\bm{\alpha}}=\alpha_{1}\,\hat{\bf e}_{1}\hat{\bf e}_{1}.$ (66)

The interaction energy of Eq. (65) then takes the form

$\displaystyle E$ $\displaystyle=$ $\displaystyle-\frac{\hbar
c\alpha_{1}\sigma_{\phi}}{32\pi^{2}}\frac{13a}{(a^{2}+h^{2})^{\frac{7}{2}}}\int_{0}^{2\pi}d\phi\,(\hat{\bf
e}_{1}\cdot\hat{\bm{\phi}})^{2}.$ (67)

Using

$\hat{\bf e}_{1}\cdot\hat{\bm{\phi}}=\sin\theta\sin(\phi_{s}-\phi)$ (68)

we have

$\int_{0}^{2\pi}d\phi\,(\hat{\bf
e}_{1}\cdot\hat{\bm{\phi}})^{2}=\pi\sin^{2}\theta.$ (69)

Thus, the interaction energy in Eq. (67) takes the form

$\displaystyle E$ $\displaystyle=$ $\displaystyle-\frac{\hbar
c\alpha_{1}\sigma_{\phi}}{32\pi}\frac{13a\sin^{2}\theta}{(a^{2}+h^{2})^{\frac{7}{2}}}.$
(70)

The orientation dependence of the polarizability $\hat{\bf e}_{1}$ with
respect to the symmetry axis $\hat{\bf z}$ is completely contained in
$\sin^{2}\theta$. This is a generic feature. The interaction energy for
permanent dipoles have an orientation dependence of the form $\cos\theta$ and
that for polarizable atoms have orientation dependence of the form $\cos
2\theta=1-2\sin^{2}\theta$. The dependence on height $h$ is monotonic, and
thus does not lead to repulsion.

####  V.1.2 Case: ${\bf\sigma}_{\phi}$ and $\alpha_{2}$

Next, let us assume the polarizability of the atom to be purely in the
direction $\hat{\bf e}_{2}$, such that,

${\bm{\alpha}}=\alpha_{2}\,\hat{\bf e}_{2}\hat{\bf e}_{2}.$ (71)

Using

$\hat{\bf
e}_{2}\cdot\hat{\bm{\phi}}=\cos\beta\cos(\phi_{s}-\phi)+\cos\theta\sin\beta\sin(\phi_{s}-\phi)$
(72)

we have

$\int_{0}^{2\pi}d\phi\,(\hat{\bf
e}_{2}\cdot\hat{\bm{\phi}})^{2}=\pi(\cos^{2}\beta+\cos^{2}\theta\sin^{2}\beta).$
(73)

Thus, the interaction energy of Eq. (65) takes the form

$\displaystyle E$ $\displaystyle=$ $\displaystyle-\frac{\hbar
c\alpha_{2}\sigma_{\phi}}{32\pi}\frac{13a(\cos^{2}\beta+\cos^{2}\theta\sin^{2}\beta)}{(a^{2}+h^{2})^{\frac{7}{2}}}.$
(74)

Note that for $\theta=0$ the interaction energy has azimuthal symmetry and is
independent of $\beta$, which is expected. Observe that for $\beta=\pi/2$ and
$\theta\to\theta-(\pi/2)$ in Eq. (74), we obtain the result for $\hat{\bf
e}_{1}$ polarization in Eq. (70). This corresponds to the fact that, in Eqs.
(61), a rotation about $\hat{\bf e}_{1}$ by an angle of $(\pi/2)-\beta$ and
then a rotation about the new $\hat{\bf e}_{2}$ by $-\pi/2$ takes the
direction of $\hat{\bf e}_{2}$ to $\hat{\bf e}_{1}$. In general the
interaction energy for an atom is a linear combination of the plausible
polarizabilities.

####  V.1.3 Case: ${\bf\sigma}_{\phi}$ and $\alpha_{3}$

Similarly, when the polarizability of the atom is purely in the direction
$\hat{\bf e}_{3}$, such that

${\bm{\alpha}}=\alpha_{3}\,\hat{\bf e}_{3}\hat{\bf e}_{3},$ (75)

using

$\hat{\bf
e}_{3}\cdot\hat{\bm{\phi}}=-\sin\beta\cos(\phi_{s}-\phi)+\cos\theta\cos\beta\sin(\phi_{s}-\phi),$
(76)

we have

$\int_{0}^{2\pi}d\phi\,(\hat{\bf
e}_{3}\cdot\hat{\bm{\phi}})^{2}=\pi(\sin^{2}\beta+\cos^{2}\theta\cos^{2}\beta).$
(77)

Thus, the interaction energy of Eq. (65) takes the form

$\displaystyle E$ $\displaystyle=$ $\displaystyle-\frac{\hbar
c\alpha_{2}\sigma_{\phi}}{32\pi}\frac{13a(\sin^{2}\beta+\cos^{2}\theta\cos^{2}\beta)}{(a^{2}+h^{2})^{\frac{7}{2}}}.$
(78)

Note that for $\theta=0$, the interaction energy has azimuthal symmetry.
Further, we observe that the interaction energy for $\hat{\bf e}_{3}$
polarization in Eq. (78) is obtained from the interaction energy for $\hat{\bf
e}_{2}$ polarization in Eq. (74) by the replacement $\beta\to\beta+(\pi/2)$.
This is true in general because $\hat{\bf e}_{2}$ and $\hat{\bf e}_{3}$ are
orthogonal vectors in the plane perpendicular to $\hat{\bf e}_{1}$, refer Eqs.
(61), and are thus related by a ninety degree rotation in angle $\beta$.

###  V.2 Radially polarizable ring

Next, let the polarizability of the ring be purely in the radial direction,

${\bm{\sigma}}=\sigma_{\rho}\,\hat{\bm{\rho}}\hat{\bm{\rho}},$ (79)

where $\hat{\bm{\rho}}$ is a unit vector normal to the ring in the $x$-$y$
plane. The expression for the interaction energy in Eq. (57) can be expressed
in the form

$\displaystyle E$ $\displaystyle=$ $\displaystyle-\frac{\hbar
c\sigma_{\rho}}{32\pi^{2}}\frac{a}{(a^{2}+h^{2})^{\frac{7}{2}}}\int_{0}^{2\pi}d\phi\Big{[}13(\hat{\bm{\rho}}\cdot{\bm{\alpha}}\cdot\hat{\bm{\rho}})$
(80) $\displaystyle\hskip 14.22636pt-56(\hat{\bf
r}\cdot{\bm{\alpha}}\cdot\hat{\bm{\rho}})(\hat{\bm{\rho}}\cdot\hat{\bf
r})+63(\hat{\bf r}\cdot{\bm{\alpha}}\cdot\hat{\bf
r})(\hat{\bm{\rho}}\cdot\hat{\bf r})^{2}\Big{]}.\hskip 14.22636pt$

Using the representation for $\hat{\bf r}$ in Eq. (56) to evaluate
$(\hat{\bm{\rho}}\cdot\hat{\bf r})$ we obtain

$\displaystyle E$ $\displaystyle=$ $\displaystyle-\frac{\hbar
c\sigma_{\rho}}{32\pi^{2}}\frac{a}{(a^{2}+h^{2})^{\frac{7}{2}}}\int_{0}^{2\pi}d\phi\bigg{[}13(\hat{\bm{\rho}}\cdot{\bm{\alpha}}\cdot\hat{\bm{\rho}})\hskip
39.83385pt$ (81) $\displaystyle\hskip 42.67912pt-\frac{56a(\hat{\bf
r}\cdot{\bm{\alpha}}\cdot\hat{\bm{\rho}})}{\sqrt{a^{2}+h^{2}}}+\frac{63a^{2}(\hat{\bf
r}\cdot{\bm{\alpha}}\cdot\hat{\bf r})}{(a^{2}+h^{2})}\bigg{]}.\hskip
14.22636pt$

####  V.2.1 Case: ${\bf\sigma}_{\rho}$ and $\alpha_{1}$

Let

${\bm{\alpha}}=\alpha_{1}\,\hat{\bf e}_{1}\hat{\bf e}_{1}.$ (82)

Now the relevant integrals involved are

$\int_{0}^{2\pi}d\phi\,(\hat{\bf r}\cdot\hat{\bf e}_{1})(\hat{\bf
e}_{1}\cdot\hat{\bm{\rho}})=\frac{\pi a\sin^{2}\theta}{\sqrt{a^{2}+h^{2}}}$
(83)

and

$\int_{0}^{2\pi}d\phi\,(\hat{\bf r}\cdot\hat{\bf
e}_{1})^{2}=\frac{\pi(a^{2}\sin^{2}\theta+2h^{2}\cos^{2}\theta)}{(a^{2}+h^{2})}.$
(84)

Using these integrals the interaction energy in Eq. (81) can be expressed in
the form

$\displaystyle E$ $\displaystyle=$ $\displaystyle-\frac{\hbar
c\alpha_{1}\sigma_{\rho}}{32\pi}\frac{a}{(a^{2}+h^{2})^{\frac{11}{2}}}\Big{[}126h^{2}a^{2}\cos^{2}\theta$
(85) $\displaystyle\hskip
28.45274pt+(20a^{4}-30h^{2}a^{2}+13h^{4})\sin^{2}\theta\Big{]}.\hskip
14.22636pt$

This form for the interaction energy is useful to compare with other
polarizabilities for the atom. Another suitable expression for interaction
energy in Eq. (81), obtained using half-angle formulas for the trigonometric
functions in Eq. (85), is

$\displaystyle E$ $\displaystyle=$ $\displaystyle-\frac{\hbar
c\alpha_{1}\sigma_{\rho}}{64\pi}\frac{a}{(a^{2}+h^{2})^{\frac{11}{2}}}\Big{[}(20a^{4}+96h^{2}a^{2}+13h^{4})$
(86) $\displaystyle\hskip 28.45274pt-(20a^{4}-156h^{2}a^{2}+13h^{4})\cos
2\theta\Big{]}.$

Figure 7: The interaction energy between an atom with polarizability
${\bm{\alpha}}=\alpha_{1}\,\hat{\bf e}_{1}\hat{\bf e}_{1}$ and a ring of
polarizability ${\bm{\sigma}}=\sigma_{\rho}\,\hat{\bm{\rho}}\hat{\bm{\rho}}$
is illustrated in bottom left corner of figure. The energy in Eq. (86) is
plotted as function of height $h$ for different orientations of $\hat{\bf
e}_{1}$ with respect to the symmetry axis of the ring. The first repulsion
occurs for $h\ll a$ when $\theta<43.43^{\circ}$. The inset provides a zoomed
in view of the non-monotonicity, second region where the switch from
attraction to repulsion occurs, which is not visible otherwise. Repulsive
force experienced by the atom in the direction of the symmetry axis will be a
manifestation of negative slopes in these plots.

Figure 7 is a plot of interaction energy in Eq. (86) as a function of height
$h$ for different orientations $\theta$. The force of repulsion is the
manifestation of energy trying to attain a minimum value, which is present
whenever the energy plots have negative slopes. The regions with negative
slope are orientation dependent, suggesting that anisotropy in the atom’s
polarizability is important.

Let us investigate the orientation dependence of the interaction energy in Eq.
(86). There exists a point at $h\sim 0.36a$ in Fig 7 where all the curves for
different $\theta$ intersect. When the domain $h$ of the plot is extended
further we encounter another such point at $h\sim 3.45a$, which has not been
captured in Fig 7. These points for height $h$ are determined by the zeros of
the polynomial that is coefficient of $\cos 2\theta$ in Eq. (86). That is,

$13h^{4}-156h^{2}a^{2}+20a^{4}=0,$ (87)

which have solutions

$h=\pm a\sqrt{\frac{78\pm 8\sqrt{91}}{13}},$ (88)

or, $h=\pm 0.36\,a$ and $h=\pm 3.45\,a$. For heights below the first torsion-
free point, $h=0.36\,a$, the atom tries to orient itself vertically
($\theta=90^{\circ}$), because the energy is minimum for this orientation,
then, for heights in between the two torsion-free points, in the range
$0.36\,a<h<3.45\,a$, the atom tries to orient itself horizontally
$(\theta=0$), and for heights beyond the second torsion-free point, beyond
$3.45\,a<h$, the atom tries to orient itself vertically. This analysis
suggests sudden transitions between orientations at torsion-free points
without expense in energy, and could have applications in experiments
involving a beam of polarizable atoms; for example, a beam of helium dimer. We
shall postpone discussions of such applications to another occasion.

Figure 8: The shaded regions in the parameter space of height $h$ and
orientation angle $\theta$ above leads to repulsion between an anisotropic
atom with polarizability ${\bm{\alpha}}=\alpha_{1}\,\hat{\bf e}_{1}\hat{\bf
e}_{1}$ interacting with a ring of polarizability
${\bm{\sigma}}=\sigma_{\rho}\,\hat{\bm{\rho}}\hat{\bm{\rho}}$, as illustrated
in top right corner of figure and determined by Eq. (92).

The negative derivative of the energy in Eq. (86) with respect to height $h$
is the component of the force on the atom in the vertical direction, given by

$\displaystyle F$ $\displaystyle=$ $\displaystyle-\frac{\hbar
c\alpha_{1}\sigma_{\rho}}{64\pi}\frac{ha}{(a^{2}+h^{2})^{\frac{13}{2}}}\Big{[}(28a^{4}+812h^{2}a^{2}+91h^{4})$
(89) $\displaystyle\hskip 28.45274pt-(532a^{4}-1456h^{2}a^{2}+91h^{4})\cos
2\theta\Big{]}.$

For repulsion to occur the force on the atom must be positive. So, the
condition for repulsion is established using the inequality

$\displaystyle(28a^{4}+812h^{2}a^{2}+91h^{4})$ $\displaystyle\hskip
28.45274pt-(532a^{4}-1456h^{2}a^{2}+91h^{4})\cos 2\theta<0.$ (90)

In the limit when the atom is very close to the center of the ring, only the
$h\neq 0$ terms contribute. The atom experiences repulsion at the center of
the ring when

$\cos 2\theta>\frac{28}{512},$ (91)

which amounts to $\theta<\frac{1}{2}\cos^{-1}(\frac{1}{19})\sim
43.43^{\circ}$. This is consistent with plots in Fig 7. For general
orientation $\theta$, we have repulsive regions at height $h$ given by the
inequality

$\displaystyle\frac{h}{a}<\pm\sqrt{\frac{-(812+1456\cos{2\theta})\pm\sqrt{D}}{182(1-\cos{2\theta})}},$
(92)

where

$\displaystyle D$ $\displaystyle=$ $\displaystyle(812+1456\cos{2\theta})^{2}$
(93) $\displaystyle-364(1-\cos{2\theta})(28-532\cos{2\theta}).$

The solutions above are real for $D>0$. This condition allows us to find a
range of orientations for which repulsion is possible. That is, the
discriminant $D=0$ determines a condition on angle $\theta$ for repulsion.
This is given by the inequality obtained by simplifying Eq. (93),

$92+364\cos{2\theta}+273\cos^{2}{2\theta}>0.$ (94)

We find $86.98^{\circ}<\theta<93.02^{\circ}$ to be a range of orientations
allowing repulsion. Using $h=0$, we also find
$-43.43^{\circ}<\theta<43.43^{\circ}$ and $136.57<\theta<223.43^{\circ}$ to be
orientations plausible for repulsion. In Fig. 8, we plot these repulsively
interacting regions in parameter space, determined by Eq. (92).

####  V.2.2 Case: ${\bf\sigma}_{\rho}$ and $\alpha_{2}$

Let

${\bm{\alpha}}=\alpha_{2}\,\hat{\bf e}_{2}\hat{\bf e}_{2}.$ (95)

The relevant integrals involved are

$\int_{0}^{2\pi}d\phi\,(\hat{\bf r}\cdot\hat{\bf e}_{2})(\hat{\bf
e}_{2}\cdot\hat{\bm{\rho}})=\frac{\pi
a(\cos^{2}\beta+\sin^{2}\beta\cos^{2}\theta)}{\sqrt{a^{2}+h^{2}}}$ (96)

and

$\displaystyle\int_{0}^{2\pi}d\phi\,(\hat{\bf r}\cdot\hat{\bf e}_{2})^{2}$
$\displaystyle=$ $\displaystyle\frac{\pi
a^{2}(\cos^{2}\beta+\sin^{2}\beta\cos^{2}\theta)}{(a^{2}+h^{2})}$ (97)
$\displaystyle+\frac{2\pi h^{2}\sin^{2}\beta\sin^{2}\theta)}{(a^{2}+h^{2})}.$

Using these integrals, we determine the interaction energy in Eq. (81) to be

$\displaystyle E=-\frac{\hbar
c\alpha_{2}\sigma_{\rho}}{32\pi}\frac{a}{(a^{2}+h^{2})^{\frac{11}{2}}}\Big{[}126h^{2}a^{2}\sin^{2}\beta\sin^{2}\theta\hskip
28.45274pt$ $\displaystyle\hskip
8.53581pt+(20a^{4}-30h^{2}a^{2}+13h^{4})(\cos^{2}\beta+\sin^{2}\beta\cos^{2}\theta)\Big{]}.$
(98)

The expression for energy in Eq. (98) has the same properties under rotations
discussed after Eq. (74). Thus, for $\beta=\pi/2$ and
$\theta\to\theta-(\pi/2)$, the energy in Eq. (98) becomes the energy for
$\hat{\bf e}_{1}$ polarization in Eq. (85).

####  V.2.3 Case: ${\bf\sigma}_{\rho}$ and $\alpha_{3}$

Let

${\bm{\alpha}}=\alpha_{3}\,\hat{\bf e}_{3}\hat{\bf e}_{3}.$ (99)

The interaction energy for this case is given by

$\displaystyle E=-\frac{\hbar
c\alpha_{3}\sigma_{\rho}}{32\pi}\frac{a}{(a^{2}+h^{2})^{\frac{11}{2}}}\Big{[}126h^{2}a^{2}\cos^{2}\beta\sin^{2}\theta\hskip
28.45274pt$ $\displaystyle\hskip
8.53581pt+(20a^{4}-30h^{2}a^{2}+13h^{4})(\sin^{2}\beta+\cos^{2}\beta\cos^{2}\theta)\Big{]}.\hskip
22.76219pt$ (100)

This result is obtained from the case of $\hat{\bf e}_{2}$ polarization by a
rotation of ninety degrees about the $\hat{\bf e}_{3}$, which is obtained by
the replacement $\beta\to(\pi/2)-\beta$ and swapping
$\cos\beta\leftrightarrow\sin\beta$, as observed after Eq. (78).

###  V.3 Ring polarizable about its symmetry axis

Finally, let the polarizability of the ring be purely along the symmetry axis
of the ring, chosen to be $\hat{\bf z}$,

${\bm{\sigma}}=\sigma_{z}\,\hat{\bf z}\hat{\bf z}.$ (101)

The expression for the interaction energy is of the form

$\displaystyle E$ $\displaystyle=$ $\displaystyle-\frac{\hbar
c\sigma_{z}}{32\pi^{2}}\frac{a}{(a^{2}+h^{2})^{\frac{7}{2}}}\int_{0}^{2\pi}d\phi\Big{[}13(\hat{\bf
z}\cdot{\bm{\alpha}}\cdot\hat{\bf z})$ (102) $\displaystyle\hskip
14.22636pt-56(\hat{\bf r}\cdot{\bm{\alpha}}\cdot\hat{\bf z})(\hat{\bf
z}\cdot\hat{\bf r})+63(\hat{\bf r}\cdot{\bm{\alpha}}\cdot\hat{\bf r})(\hat{\bf
z}\cdot\hat{\bf r})^{2}\Big{]}.\hskip 14.22636pt$

Using Eq. (56) to evaluate $(\hat{\bf z}\cdot\hat{\bf r})$, we obtain

$\displaystyle E$ $\displaystyle=$ $\displaystyle-\frac{\hbar
c\sigma_{z}}{32\pi^{2}}\frac{a}{(a^{2}+h^{2})^{\frac{7}{2}}}\int_{0}^{2\pi}d\phi\bigg{[}13(\hat{\bf
z}\cdot{\bm{\alpha}}\cdot\hat{\bf z})\hskip 39.83385pt$ (103)
$\displaystyle\hskip 42.67912pt-\frac{56h(\hat{\bf
r}\cdot{\bm{\alpha}}\cdot\hat{\bf
z})}{\sqrt{a^{2}+h^{2}}}+\frac{63h^{2}(\hat{\bf
r}\cdot{\bm{\alpha}}\cdot\hat{\bf r})}{(a^{2}+h^{2})}\bigg{]}.\hskip
14.22636pt$

####  V.3.1 Case: ${\bf\sigma}_{z}$ and $\alpha_{1}$

Let

${\bm{\alpha}}=\alpha_{1}\,\hat{\bf e}_{1}\hat{\bf e}_{1}.$ (104)

The relevant integrals involved are

$\displaystyle\int_{0}^{2\pi}d\phi\,(\hat{\bf z}\cdot\hat{\bf e}_{1})^{2}$
$\displaystyle=$ $\displaystyle 2\pi\cos^{2}\theta,$ (105a)
$\displaystyle\int_{0}^{2\pi}d\phi\,(\hat{\bf r}\cdot\hat{\bf e}_{1})(\hat{\bf
e}_{1}\cdot\hat{\bf z})$ $\displaystyle=$ $\displaystyle-\frac{2\pi
h\cos^{2}\theta}{\sqrt{a^{2}+h^{2}}},$ (105b)
$\displaystyle\int_{0}^{2\pi}d\phi\,(\hat{\bf r}\cdot\hat{\bf e}_{1})^{2}$
$\displaystyle=$
$\displaystyle\frac{\pi(a^{2}\sin^{2}\theta+2h^{2}\cos^{2}\theta)}{(a^{2}+h^{2})}.\hskip
25.60747pt$ (105c)

Using these integrals, the interaction energy in Eq. (103) can be expressed in
the form

$\displaystyle E$ $\displaystyle=$ $\displaystyle-\frac{\hbar
c\alpha_{1}\sigma_{z}}{32\pi}\frac{a}{(a^{2}+h^{2})^{\frac{11}{2}}}\Big{[}63h^{2}a^{2}\sin^{2}\theta$
(106) $\displaystyle\hskip
28.45274pt+(26a^{4}-60h^{2}a^{2}+40h^{4})\cos^{2}\theta\Big{]}.\hskip
14.22636pt$

The differences between the interaction energy expressions for the radially
polarizable ring in Eq. (85) and axially polarizable ring in Eq. (106) are
insightful and brings out the source of separate contributions to the
interaction energy. Using half-angle formulas for the trigonometric functions
in Eq. (106), we can express the interaction energy in the form

$\displaystyle E$ $\displaystyle=$ $\displaystyle-\frac{\hbar
c\alpha_{1}\sigma_{z}}{64\pi}\frac{a}{(a^{2}+h^{2})^{\frac{11}{2}}}\Big{[}(26a^{4}+3h^{2}a^{2}+40h^{4})$
(107) $\displaystyle\hskip 28.45274pt+(26a^{4}-123h^{2}a^{2}+40h^{4})\cos
2\theta\Big{]}.$

Figure 9: The interaction energy of Eq. (107) is plotted as function of height
$h$ for different orientations $\theta$. The atom has polarizability
${\bm{\alpha}}=\alpha_{1}\,\hat{\bf e}_{1}\hat{\bf e}_{1}$ and the ring has
polarizability ${\bm{\sigma}}=\sigma_{z}\,\hat{\bf z}\hat{\bf z}$, as
illustrated in bottom left corner of figure. The inset provides a zoomed-in
view of the non-monotonicity.

In Fig. 9, we plot the interaction energy of Eq. (107) as a function of height
$h$ for various orientations $\theta$. The general features of the plot in
Fig. 9 for the axial polarization of ring imitates that of the plot in Fig. 7
for the radial polarization of ring. The specifics of the interaction are
different, but the qualitative features are the same. A striking difference is
that for the radial polarization the energy is minimized at $h=0$ for
$\theta=90^{\circ}$, while for the axial polarization the energy is minimized
at $h=0$ for $\theta=0$. That is, for both $h=0$ and $h\to\infty$, the atom
tries to align its polarization with that of the ring.

Figure 10:  The shaded regions in the parameter space of height $h$ and
orientation angle $\theta$ above leads to repulsion between an anisotropic
atom with polarizability ${\bm{\alpha}}=\alpha_{1}\,\hat{\bf e}_{1}\hat{\bf
e}_{1}$ interacting with a ring of polarizability
${\bm{\sigma}}=\sigma_{z}\,\hat{\bf z}\hat{\bf z}$, as illustrated in top
right corner of figure.

The heights for which the interaction energy is orientation independent is
given by the zeros of

$26a^{4}-123h^{2}a^{2}+40h^{4}=0,$ (108)

with solutions

$h=\pm a\sqrt{\frac{123\pm\sqrt{10969}}{80}},$ (109)

at $h=\pm 0.48\,a$ and $h=\pm 1.69\,a$. The expression for the component of
the force along the symmetry axis is

$\displaystyle F$ $\displaystyle=$ $\displaystyle-\frac{\hbar
c\alpha_{1}\sigma_{z}}{64\pi}\frac{7ha}{(a^{2}+h^{2})^{\frac{13}{2}}}\Big{[}(40a^{4}-19h^{2}a^{2}+40h^{4})$
(110) $\displaystyle\hskip 28.45274pt+(76a^{4}-181h^{2}a^{2}+40h^{4})\cos
2\theta\Big{]}.$

The repulsive regions in height $h$ is given by the inequality

$\displaystyle\frac{h}{a}<\pm\sqrt{\frac{(19+181\cos{2\theta})\pm\sqrt{D}}{80(1+\cos{2\theta})}}$
(111)

where

$\displaystyle D$ $\displaystyle=$ $\displaystyle(19+181\cos{2\theta})^{2}$
(112) $\displaystyle-160(1+\cos{2\theta})(40+76\cos{2\theta})$
$\displaystyle=$ $\displaystyle-6039-11682\cos 2\theta+20601\cos^{2}2\theta.$

The critical angles of the orientations for which repulsion begins is
determined by $D=0$. This shows that repulsion only occurs when
$0<\theta<13.27^{\circ}$ and $166.73<\theta<180^{\circ}$. The other critical
value is determined by $h\to 0$, given by

$\cos 2\theta=-\frac{40}{76},$ (113)

which has solutions $60.88^{\circ}<\theta<119.12^{\circ}$.

####  V.3.2 Case: ${\bf\sigma}_{z}$ and $\alpha_{2}$

For

${\bm{\alpha}}=\alpha_{2}\,\hat{\bf e}_{2}\hat{\bf e}_{2},$ (114)

the interaction energy in Eq. (103) can be expressed in the form

$\displaystyle E$ $\displaystyle=$ $\displaystyle-\frac{\hbar
c\alpha_{2}\sigma_{z}}{32\pi}\frac{a}{(a^{2}+h^{2})^{\frac{11}{2}}}\Big{[}63h^{2}a^{2}(\cos^{2}\beta+\sin^{2}\beta\cos^{2}\theta)$
(115) $\displaystyle\hskip
28.45274pt+(26a^{4}-60h^{2}a^{2}+40h^{4})\sin^{2}\beta\sin^{2}\theta\Big{]}.\hskip
14.22636pt$

The orientation dependence in this energy should be compared with that for the
radial polarization in Eq. (98).

####  V.3.3 Case: ${\bf\sigma}_{z}$ and $\alpha_{3}$

For

${\bm{\alpha}}=\alpha_{3}\,\hat{\bf e}_{3}\hat{\bf e}_{3},$ (116)

the interaction energy in Eq. (103) can be expressed in the form

$\displaystyle E$ $\displaystyle=$ $\displaystyle-\frac{\hbar
c\alpha_{3}\sigma_{z}}{32\pi}\frac{a}{(a^{2}+h^{2})^{\frac{11}{2}}}\Big{[}63h^{2}a^{2}(\sin^{2}\beta+\cos^{2}\beta\cos^{2}\theta)$
(117) $\displaystyle\hskip
28.45274pt+(26a^{4}-60h^{2}a^{2}+40h^{4})\cos^{2}\beta\sin^{2}\theta\Big{]}.\hskip
14.22636pt$

Again, the orientation dependence in this expression should be contrasted with
that in Eq. (100).

###  V.4 Partially isotropic ring

If the polarizability of the ring is isotropic in the $x$-$y$ plane we have

${\bm{\sigma}}=\sigma\,(\hat{\bm{\rho}}\hat{\bm{\rho}}+\hat{\bm{\phi}}\hat{\bm{\phi}}).$
(118)

The interaction energy for this case will be a linear sum of the individual
energies for the radially and tangentially polarizable cases. If the
polarizability of the atom is of the form

${\bm{\alpha}}=\alpha_{1}\,\hat{\bf e}_{1}\hat{\bf e}_{1},$ (119)

the interaction energy is obtained by adding the energies in Eqs. (70) and
(86).

$\displaystyle E$ $\displaystyle=$ $\displaystyle-\frac{\hbar
c\alpha_{1}\sigma}{64\pi}\frac{a}{(a^{2}+h^{2})^{\frac{11}{2}}}\Big{[}(33a^{4}+122h^{2}a^{2}+26h^{4})$
(120) $\displaystyle\hskip 28.45274pt-(33a^{4}-130h^{2}a^{2}+26h^{4})\cos
2\theta\Big{]}.$

We see that the qualitative features do not change relative to that of a
radially polarizable ring. It is of interest to inquire how the features of
energy plot change from a ring to that of a plate. To this end, we study a
polarizable annular disc.

##  VI Polarizable atom above a dielectric annular disc

Consider a polarizable annular disc of inner radius $a$ and outer radius $b$,
with polarizability described by

${\bm{\chi}}={\bm{\lambda}}\delta(z)\theta(\rho-a)\theta(b-\rho).$ (121)

In the limit $b\to\infty$ this corresponds to a plate of infinite extent with
a circular aperture of radius $a$. In the limit $a\to 0$ this corresponds to
circular disc of radius $b$. In the simple limit $b\to a$ the annular disc is
of course non existent. However, the delicate limit of $b\to a$ in conjunction
with $\lambda\to\infty$ such that

$\sigma=\lambda(b-a)$ (122)

is kept fixed, leads to a construction of an infinitely thin ring, after
recognizing the $\delta$-function representation in terms of step functions

$\delta(\rho-a)=\lim_{b\to a}\frac{\theta(b-\rho)\theta(\rho-a)}{(b-a)}.$
(123)

In this manner, we can obtain the results for ring, and that for plate with
circular aperture, from the results for an annular disc.

###  VI.1 Partially isotropic annular disc

Let the polarizability of the atom be of the form

${\bm{\alpha}}=\alpha_{1}\,\hat{\bf e}_{1}\hat{\bf e}_{1}.$ (124)

Let the annular disc be isotropically polarizable in the $x$-$y$ plane,

${\bm{\lambda}}=\lambda{\bf 1}_{\perp}=\lambda(\hat{\bf x}\hat{\bf x}+\hat{\bf
y}\hat{\bf
y})=\lambda(\hat{\bm{\rho}}\hat{\bm{\rho}}+\hat{\bm{\phi}}\hat{\bm{\phi}}).$
(125)

Then, the expression for the interaction energy in Eq. (50) takes the
simplified form

$\displaystyle E$ $\displaystyle=$ $\displaystyle-\frac{\hbar
c\alpha_{1}\lambda}{32\pi^{2}}\int_{a}^{b}d\rho\frac{\rho}{(\rho^{2}+h^{2})^{\frac{7}{2}}}\int_{0}^{2\pi}d\phi\Big{[}13(\hat{\bf
e}_{1}\cdot{\bf 1}_{\perp}\cdot\hat{\bf e}_{1})$ (126)
$\displaystyle-56\,(\hat{\bf r}\cdot\hat{\bf e}_{1})(\hat{\bf e}_{1}\cdot{\bf
1}_{\perp}\cdot\hat{\bf r})+63\,(\hat{\bf r}\cdot\hat{\bf e}_{1})^{2}(\hat{\bf
r}\cdot{\bf 1}_{\perp}\cdot\hat{\bf r})\Big{]}.\hskip 22.76219pt$

Completing the integral in $\phi$ we obtain

$\displaystyle E$ $\displaystyle=$ $\displaystyle-\frac{\hbar
c\alpha_{1}\lambda}{64\pi}\int_{a}^{b}d\rho\frac{\rho}{(\rho^{2}+h^{2})^{\frac{11}{2}}}\Big{[}(33\rho^{4}+122h^{2}\rho^{2}+26h^{4})$
(127) $\displaystyle\hskip
28.45274pt-(33\rho^{4}-130h^{2}\rho^{2}+26h^{4})\cos 2\theta\Big{]}.$

In the limit $\lambda\to\infty$ and $b\to a$ such that $\lambda(b-a)=\sigma$
we have $\lambda\int_{a}^{b}d\rho\to\sigma$, which then leads to the
interaction energy for a partially isotropic ring in Eq. (120). Completing the
integral in $\rho$ Eq. (127) leads to

$\displaystyle E$ $\displaystyle=$ $\displaystyle-\frac{\hbar
c\alpha_{1}\lambda}{320\pi}\frac{(-1)}{(\rho^{2}+h^{2})^{\frac{9}{2}}}\Big{[}(33\rho^{4}+106h^{2}\rho^{2}+38h^{4})$
(128) $\displaystyle\hskip 28.45274pt-(33\rho^{4}-74h^{2}\rho^{2}-2h^{4})\cos
2\theta\Big{]}\bigg{|}^{\rho=b}_{\rho=a}.$

In the limit $b\to\infty$, this reproduces the expression for the energy of an
infinite plate with a circular aperture in Eq. (45).

The characteristic features of the interaction energy for an annular disc in
Eq. (128) are similar to those of a partially isotropic ring in Eq. (120).
These are, the two orientation independent torsion free heights on each side
of the disc, and the orientation preferences at $h=0$ and $h\to\infty$. The
orientation independent heights are determined by setting the coefficient of
the term with $\cos 2\theta$ to zero in Eq. (128). That is,

$\frac{(33b^{4}-74b^{2}h^{2}-2h^{4})}{(b^{2}+h^{2})^{\frac{9}{2}}}-\frac{(33a^{4}-74a^{2}h^{2}-2h^{4})}{(a^{2}+h^{2})^{\frac{9}{2}}}=0,$
(129)

which has four solutions for $h$. Two of the solutions are above the disc
($h>0$), and the other two are opposite in sign and symmetrically below the
disc. Denoting the positive solutions as $h_{1}$ and $h_{2}$, we find

$0.52a\xleftarrow[\text{ring}]{a\leftarrow
b}h_{1}\xrightarrow[\text{plate}]{b\to\infty}0.66a$ (130a) and
$2.18a\xleftarrow[\text{ring}]{a\leftarrow
b}h_{2}\xrightarrow[\text{plate}]{b\to\infty}\infty,$ (130b)

such that $h_{1}$ and $h_{2}$ vary monotonically. At $h=0$ the orientation
preference of the atom is to direct its polarizability in the plane of the
plate ($\theta=90^{\circ}$). As the atom is moved away from this position
along the axis, the orientation preference abruptly switches to $\theta=0$ as
the atom crosses the height $h=h_{1}$. Afterwards it switches back to
$\theta=90^{\circ}$ when we cross the height $h=h_{2}$. The existence of two
orientation independent heights, on each side of disc, thus guarantees that
the orientation preference at $h=0$ will be the same as the orientation
preference at $h\to\infty$. Conversely, since the plate ($b\to\infty$) has
only one orientation independent height, the other being at infinity, see Eq.
(130b), the implication is that orientation preference of the atom at $h=0$ is
orthogonal to that at $h\to\infty$.

###  VI.2 Radially polarizable annular disc

For a polarizable atom on the symmetry axis of an annular disc that is
polarizable radially,

${\bm{\lambda}}=\lambda_{\rho}\hat{\bm{\rho}}\hat{\bm{\rho}},$ (131)

the interaction energy is

$\displaystyle E$ $\displaystyle=$ $\displaystyle-\frac{\hbar
c\alpha_{1}\lambda_{\rho}}{64\pi}\frac{(-1)}{(\rho^{2}+h^{2})^{\frac{9}{2}}}\Big{[}(4\rho^{4}+16h^{2}\rho^{2}+5h^{4})$
(132) $\displaystyle\hskip 28.45274pt-(4\rho^{4}-20h^{2}\rho^{2}-3h^{4})\cos
2\theta\Big{]}\bigg{|}^{\rho=b}_{\rho=a}.$

This expression tends to the expression for a ring in Eq. (86), and to the
expression for a plate in Eq. (47), in the limits $b\to a$ and $b\to\infty$,
respectively.

The characteristic features of the interaction energy in Eq. (132) for
radially polarizable annular disc qualitatively mimics that of partially
isotropically polarizable annular disc in Eq. (128). The orientation
independent heights are now determined by

$\frac{(4b^{4}-20b^{2}h^{2}-3h^{4})}{(b^{2}+h^{2})^{\frac{9}{2}}}-\frac{(4a^{4}-20a^{2}h^{2}-3h^{4})}{(a^{2}+h^{2})^{\frac{9}{2}}}=0,$
(133)

which again has four solutions. Two of the solutions above the disc, ($h_{1}$
and $h_{2}$, with $0<h_{1}<h_{2}$), are

$0.36a\xleftarrow[\text{ring}]{a\leftarrow
b}h_{1}\xrightarrow[\text{plate}]{b\to\infty}0.44a$ (134a) and
$3.45a\xleftarrow[\text{ring}]{a\leftarrow
b}h_{2}\xrightarrow[\text{plate}]{b\to\infty}\infty.$ (134b)

Again, the existence of two positive orientation independent heights for the
case of annular disc, and for the limiting case of disc becoming a ring,
implies that the orientation preference of the atom at $h=0$ and $h\to\infty$
is the same. In the limiting case of an annular disc becoming an annular plate
one of the orientation independent heights moves to infinity, which implies
that the orientation preference at $h=0$ is orthogonal to that at $h\to\infty$
for an annular plate. These points have been illustrated in Fig. 11.

Orientation |  | Criteria for second region of repulsion
---|---|---
$\theta=90^{\circ}$ |  |  $a<b<1.257a$
$\theta=88.2^{\circ}$ |  | $a<b<1.201a$
$\theta=87.3^{\circ}$ |  | $a<b<1.107a$
Table 1: This table lists the criteria when the region of repulsion for
intermediate distances can occur between an anisotropically polarizable
annular disc and anisotropically polarizable atom for certain orientations.
The left column gives the orientation of the atom deviating away from the
symmetry axis when the disc is polarizable in the radial direction. The right
column gives the criteria for second region of repulsion to be present. The
corresponding chart for axially polarizable annular disc has been presented in
Ref. Marchetta  _et al._ (2020).

A motivation for analyzing the disc was to determine when does the second
region of repulsion emerge in the transition of the annular disc shrinking
down to a ring. These values have been evaluated numerically and are listed in
Table 1.

###  VI.3 Axially polarizable annular disc

When the annular disc is polarizable along the direction of the symmetry axis
of the disc,

${\bm{\lambda}}=\lambda_{z}\hat{\bf z}\hat{\bf z},$ (135)

the interaction energy is

$\displaystyle E$ $\displaystyle=$ $\displaystyle-\frac{\hbar
c\alpha_{1}\lambda_{z}}{64\pi}\frac{1}{5}\frac{(-1)}{(\rho^{2}+h^{2})^{\frac{9}{2}}}\Big{[}(26\rho^{4}+17h^{2}\rho^{2}+26h^{4})$
(136) $\displaystyle\hskip 28.45274pt+(26\rho^{4}-73h^{2}\rho^{2}+6h^{4})\cos
2\theta\Big{]}\bigg{|}^{\rho=b}_{\rho=a}.$

The expression for a ring in Eq. (107), and the expression for a plate in Eq.
(49), are reproduced from the interaction energy of Eq. (136) in the limits
$b\to a$ and $b\to\infty$, respectively.

The orientation independent heights are now determined by

$\frac{(26b^{4}-73b^{2}h^{2}+6h^{4})}{(b^{2}+h^{2})^{\frac{9}{2}}}-\frac{(26a^{4}-73a^{2}h^{2}+6h^{4})}{(a^{2}+h^{2})^{\frac{9}{2}}}=0,$
(137)

with the solutions above the disc satisfying

$0.48a\xleftarrow[\text{ring}]{a\leftarrow
b}h_{1}\xrightarrow[\text{plate}]{b\to\infty}0.60a$ (138a) and
$1.69a\xleftarrow[\text{ring}]{a\leftarrow
b}h_{2}\xrightarrow[\text{plate}]{b\to\infty}3.44a.$ (138b)

Unlike the case of radial polarizability, here, two orientation independent
heights on each side of disc exists even in the limit when the disc becomes a
plate. That is, the second orientation independent height does not move away
to infinity in the limiting case of a plate. This implies that the orientation
preference of the atom at $h=0$ and $h\to\infty$ is the same. See illustration
in Fig. 11. This should be contrasted with that of radially polarizable disc
in which case the orientation preference of the polarizability of the atom at
$h=0$ is orthogonal to that at $h\to\infty$.

##  VII Summary of results

Figure 11: Difference in interaction energy $\Delta E$ between $\theta=0$ and
$\theta=\pi/2$ plotted as a function of height $h$. The intersection of the
energy curves with abscissa represent orientation independent points, marked
$h_{1}$ and $h_{2}$. The orientation dependence at each height is shown as
double-arrowed vectors adjacent to the horizontal axis. Observe that the
orientation preferences for the plate is different for radial and axial
polarizabilities.

Admittedly, we have considered too many cases and it has been hard to keep
track. A brief summary of our results in this paper has been presented in Ref.
Marchetta  _et al._ (2020). We already pointed out that the qualitative
features of isotropically polarizability in a plane is identical to that of
radially polarizable case. We have also emphasized that the characteristic
features of the interaction energy are the orientation independent heights and
the orientation preferences at $h=0$ and $h=\infty$. To this end, we feel it
is sufficient to bring out the difference between the radially polarizable
case and the axially polarizable case.

We summarize the limiting forms of the expressions for interaction energy for
radial polarization as

$\displaystyle{\bm{\lambda}}=\lambda\hat{\bm{\rho}}\hat{\bm{\rho}}:\qquad\text{Eq.\,(\ref{CPannD-e1lr-
apr})}\begin{cases}\xrightarrow[\text{ring}]{b\to
a}\text{Eq.\,(\ref{intE-e1lr})}\\\\[19.91692pt]
\xrightarrow[\text{plate}]{b\to\infty}\text{Eq.\,(\ref{energy-atom-plate-hole-
lr})}\end{cases}$ (139a) and Eq. (86)
$\displaystyle\begin{cases}\xrightarrow{h\to 0}-E_{r}20(1-\cos
2\theta),\\\\[8.53581pt]
\xrightarrow{h\to\infty}-E_{r}\dfrac{a^{7}}{h^{7}}13(1-\cos
2\theta),\end{cases}$ (139b) Eq. (47)
$\displaystyle\begin{cases}\xrightarrow{h\to 0}-E_{p}4(1-\cos
2\theta),\\\\[8.53581pt]
\xrightarrow{h\to\infty}-E_{p}\dfrac{a^{5}}{h^{5}}(5+3\cos
2\theta),\end{cases}$ (139c)

where

$E_{r}=\frac{\hbar c\alpha_{1}\sigma_{\rho}}{64\pi a^{6}},\qquad
E_{p}=\frac{\hbar c\alpha_{1}\lambda_{\rho}}{64\pi a^{5}}.$ (140)

The difference in the orientation dependence between the plate and the ring
arises because minimum energy for a ring at $h\to\infty$ is decided by

$-(1-\cos 2\theta)=\begin{cases}0,&\theta=0,\\\
-2,&\theta=90^{\circ},\end{cases}$ (141)

and the minimum energy for a plate at $h\to\infty$ is decided by

$-(5+3\cos 2\theta)=\begin{cases}-8,&\theta=0,\\\
-2,&\theta=90^{\circ}.\end{cases}$ (142)

The source of this, again, can be attributed to the dyadic-dyadic interaction.

We repeat the summary for axial polarizability as

$\displaystyle{\bm{\lambda}}=\lambda\hat{\bf z}\hat{\bf
z}:\qquad\text{Eq.\,(\ref{CPannD-e1lz-
apr})}\begin{cases}\xrightarrow[\text{ring}]{b\to
a}\text{Eq.\,(\ref{intE-e1lz})}\\\\[19.91692pt]
\xrightarrow[\text{plate}]{b\to\infty}\text{Eq.\,(\ref{energy-atom-plate-hole-
lz})}\end{cases}$ (143a) and Eq. (107)
$\displaystyle\begin{cases}\xrightarrow{h\to 0}-E_{r}26(1+\cos
2\theta),\\\\[8.53581pt]
\xrightarrow{h\to\infty}-E_{r}\dfrac{a^{7}}{h^{7}}40(1+\cos
2\theta),\end{cases}$ (143b) Eq. (49)
$\displaystyle\begin{cases}\xrightarrow{h\to 0}-\dfrac{E_{p}}{5}26(1+\cos
2\theta),\\\\[8.53581pt]
\xrightarrow{h\to\infty}-\dfrac{E_{p}}{5}\dfrac{a^{5}}{h^{5}}(26+6\cos
2\theta),\end{cases}$ (143c)

where $E_{r}$ and $E_{p}$ are now given by Eq. (140) after swapping the
subscripts $\rho\to z$. Identical orientation dependence between the plate and
the ring in this case arises because minimum energy for a ring at $h\to\infty$
is decided by

$-40(1+\cos 2\theta)=\begin{cases}-80,&\theta=0,\\\
0,&\theta=90^{\circ},\end{cases}$ (144)

and the minimum energy for a plate at $h\to\infty$ is decided by

$-(26+6\cos 2\theta)=\begin{cases}-32,&\theta=0,\\\
-20,&\theta=90^{\circ}.\end{cases}$ (145)

The source of this can be attributed to the dyadic-dyadic interaction, which
is short of an intuitive understanding.

We plotted the energy difference between orientations at $\theta=0^{\circ}$
and $\theta=90^{\circ}$ to highlight the orientation independent heights and
the orientation dependence with respect to heights in Fig. 11. Among the four
plots, note that for the case of radially polarizable plate, the atom tends to
orient itself perpendicular to the direction of polarizability of the plate
for $h\to\infty$. This contrasting feature is the key result of our analysis.
We envision a beam of polarizable atoms passing through an aperture in a
dielectric plate as an application that will exploit our result.

##  VIII Casimir machine

A rotaxane Anelli  _et al._ (1991) is a molecular structure consisting of a
ring shaped molecule free to move on another dumbbell shaped molecule. A
rotaxane is a machine because the position of the ring can be controlled by
expending chemical or electrostatic energy to change the associated
interaction energy between the ‘ring’ and the ‘dumbbell’. These structures
have been studied with considerable interest to find applications in
constructing molecular machines Balzani  _et al._ (2008). Using results from
Sec. V.3.1, we propose a prototype for a rotaxane-like machine using the
configurations studied there. An illustrative diagram of such a machine is
shown in Fig. 12. Casimir-Polder approximation typically require the distances
to be in the ballpark of 100 nm, and thus our discussion here is applicable
for plausible applications in nanotechnology. The proposal here is superficial
and presented for the sake of captivating interest, in the sense that careful
details in the specific design has not been attended.

Figure 12: A prototype of a Casimir machine. Controlling the orientation
$\theta$ of the polarizability of the nanoparticle allows the movement of the
polarizable ring along the symmetry axis. For angle $\theta=0$ it is
energetically favorable for the nanoparticle to be at the center of the ring,
and for $\theta=90^{\circ}$ it is repelled from the ring. The plot is the
interaction energy of the nanoparticle and ring as a function of height $h$
for $\theta=0$ and $\theta=90^{\circ}$. The configuration states, $A$, $B$,
$C$, and $D$, in the Casimir machine are marked on the plots and illustrated
in the insets.

We consider a polarizable ring and a polarizable atom with interaction energy
given by Eq. (107) rewritten here in the form

$\displaystyle E(h,\theta)$ $\displaystyle=$
$\displaystyle-\frac{E_{0}}{52}\frac{a^{7}}{(a^{2}+h^{2})^{\frac{11}{2}}}\Big{[}(26a^{4}+3h^{2}a^{2}+40h^{4})\hskip
8.53581pt$ (146) $\displaystyle\hskip
14.22636pt+(26a^{4}-123h^{2}a^{2}+40h^{4})\cos 2\theta\Big{]},$

where $E_{0}=13\hbar c\alpha\sigma/(16\pi a^{6})$. This energy has been
plotted with respect to height $h$ in Fig. 12 for $\theta=0$ and
$\theta=90^{\circ}$. Let us assume that the atom is constrained to move on the
axis. The force between the atom and ring is given by

$F=-\frac{\partial E}{\partial h}$ (147)

and can be expressed in the form

$\displaystyle F(h,\theta)$ $\displaystyle=$
$\displaystyle-\frac{F_{0}}{52}\frac{7a^{8}h}{(a^{2}+h^{2})^{\frac{13}{2}}}\Big{[}(40a^{4}-19h^{2}a^{2}+40h^{4})\hskip
8.53581pt$ (148) $\displaystyle\hskip
14.22636pt+(76a^{4}-181h^{2}a^{2}+40h^{4})\cos 2\theta\Big{]},$

where $F_{0}=E_{0}/a$. In Fig. 13, we plot the force as a function of height
$h$. The torque on the atom is given by

$\tau=-\frac{\partial E}{\partial\theta}$ (149)

and can be expressed in the form

$\displaystyle\tau(h,\theta)$ $\displaystyle=$
$\displaystyle-\frac{E_{0}}{26}\frac{a^{7}(26a^{4}-123h^{2}a^{2}+40h^{4})\sin
2\theta}{(a^{2}+h^{2})^{\frac{11}{2}}}.\hskip 19.91692pt$ (150)

In Fig. 14, we plot the torque as a function of angle $\theta$.

Figure 13: Force verses height $h$ for $\theta=0$ and $\theta=90^{\circ}$.
States $A$, $B$, $C$, and $D$, for the Casimir machine are marked.  Figure 14:
Torque verses orientation $\theta$ for $h=0$ and $h=h_{e}$. States $A$, $B$,
$C$, and $D$, for the Casimir machine are marked.

Let us now examine the following mechanism. Consider an initial configuration
$(h=0,\theta=0)$ denoted by state ‘A’ in Figs. 12 to 14. We assume that the
atom can manipulate the orientation $\theta$ from $0^{\circ}$ to $90^{\circ}$
at the expense of internal energy. The new state $(h=0,\theta=90^{\circ})$ is
denoted by state ‘B’ in Figs. 12 to 14. This will propel the ring away due to
repulsion, leading to the configuration $(h=h_{e},\theta=90^{\circ})$ denoted
by state ‘C’ in Figs. 12 to 14. The atom at this position manipulates the
orientation $\theta$ from $90^{\circ}$ to $0^{\circ}$ at the expense of no
internal energy, see discussion after Eq. (88). The new state
$(h=h_{e},\theta=0^{\circ})$ is denoted by state ‘D’ in Figs. 12 to 14. This
will propel the ring closer to the atom due to attraction. The cycle $A\to
B\to C\to D\to A$, see Fig. 15, thus is a machine that is able to control the
position of the ring at the expense of internal energy in the step $A\to B$.

Figure 15: Configuration states in the Casimir machine in Figs. 12 to 14.

The change in energies in the individual steps in the cycle are the following:

$\displaystyle W_{A\to B}$ $\displaystyle=$
$\displaystyle-\int_{0,h=0}^{\frac{\pi}{2}}\tau
d\theta=E(0,\frac{\pi}{2})-E(0,0)>0,$ (151a) $\displaystyle W_{B\to C}$
$\displaystyle=$
$\displaystyle-\int_{0,\theta=\frac{\pi}{2}}^{h_{e}}Fdh=E(h_{e},\frac{\pi}{2})-E(0,\frac{\pi}{2})<0,\hskip
28.45274pt$ (151b) $\displaystyle W_{C\to D}$ $\displaystyle=$
$\displaystyle-\int_{0,h=h_{e}}^{\frac{\pi}{2}}\tau
d\theta=E(h_{e},0)-E(h_{e},\frac{\pi}{2})=0,$ (151c) $\displaystyle W_{D\to
A}$ $\displaystyle=$
$\displaystyle-\int_{h_{e},\theta=0}^{0}Fdh=E(0,0)-E(h_{e},0)<0,$ (151d)

where the energies are evaluated using Eq. (146). As noted, the energy input
is in $A\to B$. The step from $C\to D$ is achieved at no cost in internal
energy.

##  IX Conclusion and outlook

We have derived and presented Casimir-Polder interaction energies between an
anisotropically polarizable atom and an anisotropically polarizable ring, and
when ring is replaced with an annular disc. Then, taking the outer radius to
infinity to obtain a plate with circular aperture. We have identified specific
configurations that lead to repulsive Casimir-Polder forces on the atom. We
observe that when the relative orientations of the polarizabilities of the
atom and the ring are perpendicular to each other the atom experiences a
repulsive force when it is very close to the center of the ring. The energy-
distance plots in Figs. 3, 7, and 9, mimics the energy-distance plot for an
elongated needle shaped conductor and a plate with an aperture in Ref. Levin
_et al._ (2010). These plots also have the qualitative features of the energy-
distance plot reported for an atom and a conducting toroid in Ref. Abrantes
_et al._ (2018). As suspected early on, these interaction energies seem to be
characterized by the interaction of the individual eigenbasis of anisotropic
polarizabilities of the objects. These dyadic-dyadic interactions are very
rich due to the various possible interactions between the principal
polarizations. To illustrate this richness, we have shown that in our energy-
distance plots for an atom and ring, in addition to repulsion at short
distances when the orientations are relatively perpendicular, we find another
region of repulsion at intermediate distances when the orientations are
relatively almost parallel.

We recognize that experimental observation of repulsion resulting from
anisotropy and perforated geometries is lacking because the force is extremely
weak. Numerical estimate of these energies is of the order of $\mu$eV, which
is small. However, the energies between conducting bodies in contrast to
dielectric materials considered in this article is expected to be larger
Venkataram  _et al._ (2020). We also note that non-monotonic Casimir forces in
the context of two interlocked corrugated geometries resembling a ‘zipper’
were proposed in Ref. Rodriguez  _et al._ (2008), which has now been realized
experimentally Tang  _et al._ (2017). These developments are expected to play
an important role in the advancement of nanoscale machines.

Figure 16:  A spinning anisotropically polarizable atom of polarizability
${\bm{\alpha}}$ at a height $h$ above a dielectric ring of radius $a$. The
atom is spinning with angular speed $\omega$ about the direction of
polarization while maintaining an angle $\theta$ with the axis of the ring.

We confined our discussion here to static configurations. This prompts the
question of whether stable equilibrium can be achieved in these configurations
when the interactions are induced by quantum vacuum fluctuations? By
generalizing Earnshaw’s theorem to include fluctuation-induced forces, the
authors of Ref. Rahi  _et al._ (2010) conclude that neutral polarizable
objects cannot be in stable equilibrium in the quantum vacuum. However, like
in electrostatics, it might still be possible to have stable equilibrium in
dynamical configurations. We intend to study if a spinning polarizable atom,
as described in Fig. 16, will precess and in the process attain dynamical
stable equilibrium. This would be a Casimir-Polder analog of the
Levitron${}^{\text{TM}}$ Berry (1996).

###### Acknowledgements.

We thank Avinash Khatri, Christian Rose, Suddarsun Shivakumar, and Preston Yun
for discussions during the research. JJM acknowledges support from the REACH
Program at Southern Illinois University–Carbondale. PP and KVS remember Martin
Schaden for collaborative assistance.

## References

  * van der Waals (1873) J. D. van der Waals,  _Over de Continuiteit van den Gas-en Vloeistoftoestand, (On the continuity of the gas and liquid state)_ , Ph.D. thesis, Universiteit Leiden (Leiden University), The Netherlands (1873).
  * Eisenschitz and London (1930) R. Eisenschitz and F. London, “Über das Vrhältnis der van der Waalsschen Kräfte zu den Homöopolaren Bindungskräften,” Z. Physik 60, 491 (1930), english translation in Hettema (2000).
  * London (1930) F. London, “Zur Theorie und Systematik der Molekularkräfte,” Z. Physik 63, 245 (1930), english translation in Hettema (2000).
  * Hettema (2000) H. Hettema,  _Quantum Chemistry: Classic Scientific Papers_ , World Scientific Series in 20th century chemistry (World Scientific, 2000).
  * Casimir and Polder (1948) H. B. G. Casimir and D. Polder, “The influence of retardation on the London-van der Waals forces,” Phys. Rev. 73, 360 (1948).
  * Casimir (1948) H. B. G. Casimir, “On the attraction between two perfectly conducting plates,” Kon. Ned. Akad. Wetensch. Proc. 51, 793 (1948).
  * Milton (2011) K. A. Milton, “Resource Letter VWCPF-1: van der Waals and Casimir-Polder forces,” Am. J. Phys. 79, 697 (2011).
  * Lifshitz (1956) E. M. Lifshitz, “The theory of molecular attractive forces between solids,” Sov. Phys. JETP 2, 73 (1956), [Translated from: Zh. Eksp. Teor. Fiz. 29, 94 (1956)].
  * Dzyaloshinskii  _et al._ (1961) I. E. Dzyaloshinskii, E. M. Lifshitz, and L. P. Pitaevskii, “General theory of van der Waals’ forces,” Soviet Physics Uspekhi 4, 153 (1961), [Translated from: Usp. Fiz. Nauk 73, 381 (1961)].
  * Axilrod and Teller (1943) B. M. Axilrod and E. Teller, “Interaction of the van der Waals type between three atoms,” J. Chem. Phys. 11, 299 (1943).
  * Muto (1943) Y. Muto, J. Phys. Math. Soc. Japan 17, 629 (1943).
  * Craig and Power (1969a) D. P. Craig and E. A. Power, “The asymptotic Casimir-Polder potential for anisotropic molecules,” Chem. Phys. Lett. 3, 195 (1969a).
  * Craig and Power (1969b) D. P. Craig and E. A. Power, “The asymptotic Casimir-Polder potential from second-order perturbation theory and its generalization for anisotropic polarizabilities,” Int. J. Quantum Chem. 3, 903 (1969b).
  * Babb (2005) J. F. Babb, “Long-range atom-surface interactions for cold atoms,” J. Phys. Conf. Ser. 19, 1 (2005).
  * Levin  _et al._ (2010) M. Levin, A. P. McCauley, A. W. Rodriguez, M. T. H. Reid, and S. G. Johnson, “Casimir repulsion between metallic objects in vacuum,” Phys. Rev. Lett. 105, 090403 (2010).
  * Levin and Johnson (2011) M. Levin and S. G. Johnson, “Is the electrostatic force between a point charge and a neutral metallic object always attractive?” Am. J. Phys. 79, 843 (2011).
  * McCauley  _et al._ (2011) A. P. McCauley, A. W. Rodriguez, M. T. Homer Reid, and S. G. Johnson, “Casimir repulsion beyond the dipole regime,” (2011), arXiv:1105.0404 [quant-ph] .
  * Eberlein and Zietal (2011) C. Eberlein and R. Zietal, “Casimir-Polder interaction between a polarizable particle and a plate with a hole,” Phys. Rev. A 83, 052514 (2011).
  * Abrantes  _et al._ (2018) P. P. Abrantes, Yuri França, F. S. S. da Rosa, C. Farina, and R. de Melo e Souza, “Repulsive van der Waals interaction between a quantum particle and a conducting toroid,” Phys. Rev. A 98, 012511 (2018).
  * Shajesh and Schaden (2012) K. V. Shajesh and M. Schaden, “Repulsive long-range forces between anisotropic atoms and dielectrics,” Phys. Rev. A 85, 012523 (2012).
  * Milton  _et al._ (2012a) K. A. Milton, P. Parashar, N. Pourtolami, and I. Brevik, “Casimir-Polder repulsion: Polarizable atoms cylinders, spheres, and ellipsoids,” Phys. Rev. D 85, 025008 (2012a).
  * Milton  _et al._ (2011) K. A. Milton, E. K. Abalo, P. Parashar, N. Pourtolami, I. Brevik, and S. Å. Ellingsen, “Casimir-Polder repulsion near edges: Wedge apex and a screen with an aperture,” Phys. Rev. A 83, 062507 (2011).
  * Milton  _et al._ (2012b) K. A. Milton, E. K. Abalo, P. Parashar, N. Pourtolami, I. Brevik, and S. Å. Ellingsen, “Repulsive Casimir and Casimir–Polder forces,” J. Phys. A: Mathematical and Theoretical 45, 374006 (2012b).
  * Shajesh  _et al._ (2017) K. V. Shajesh, P. Parashar, and I. Brevik, “Casimir-Polder energy for axially symmetric systems,” Ann. Phys. 387, 166 (2017).
  * Marchetta  _et al._ (2020) J. J. Marchetta, P. Parashar, and K. V. Shajesh, “Geometrical dependence in casimir-polder repulsion,” (2020).
  * Feinberg and Sucher (1968) G. Feinberg and J. Sucher, “General form of the retarded van der Waals potential,” J. Chem. Phys. 48, 3333 (1968).
  * Boyer (1968) T. H. Boyer, “Quantum electromagnetic zero point energy of a conducting spherical shell and the Casimir model for a charged particle,” Phys. Rev. 174, 1764 (1968).
  * Shajesh and Schaden (2011) K. V. Shajesh and M. Schaden, “Many-body contributions to Green’s functions and Casimir energies,” Phys. Rev. D 83, 125032 (2011).
  * Anelli  _et al._ (1991) P. L. Anelli, N. Spencer, and J. F. Stoddart, “A molecular shuttle,” J. Am. Chem. Soc. 113, 5131 (1991).
  * Balzani  _et al._ (2008) V. Balzani, A. Credi, and M. Venturi,  _Molecular Devices and Machines: Concepts and Perspectives for the Nanoworld_ , 2nd ed. (Wiley-VCH, Weinheim, 2008).
  * Venkataram  _et al._ (2020) P. S. Venkataram, S. Molesky, P. Chao, and A. W. Rodriguez, “Fundamental limits to attractive and repulsive Casimir-Polder forces,” Phys. Rev. A 101, 052115 (2020).
  * Rodriguez  _et al._ (2008) A. W. Rodriguez, J. D. Joannopoulos, and S. G. Johnson, “Repulsive and attractive Casimir forces in a glide-symmetric geometry,” Phys. Rev. A 77, 062107 (2008).
  * Tang  _et al._ (2017) L.a Tang, M. Wang, C. Y. Ng, M. Nikolic, C. T. Chan, A. W. Rodriguez, and H. B. Chan, “Measurement of non-monotonic Casimir forces between silicon nanostructures,” Nature Photonics 11, 97 (2017).
  * Rahi  _et al._ (2010) S. J. Rahi, M. Kardar, and T. Emig, “Constraints on Stable Equilibria with Fluctuation-Induced (Casimir) Forces,” Phys. Rev. Lett. 105, 070404 (2010).
  * Berry (1996) M. V. Berry, “The Levitron${}^{\text{TM}}$: an adiabatic trap for spins,” Proc. R. Soc. Lond. A. 452, 1207 (1996).