Method for the Casimir effect in the sphere-plane geometry 

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The classical vacuum

In the history of sciences, the concept of vacuum has often been a matter of debate [1, 2], mainly because of its philosophical, religious, and metaphysical implications. Already in the beginning of the 5th century BCE, the atomists Leucippus and Democritus hold that everything is composed by atoms, and that between them lies the vacuum (or nothingness), in which movement takes place [3, 4]. In the same century, Parmenides defines the vacuum as a lack of being. Using an ontological argument, he shows that vacuum cannot exist [5, 6]. Plato adds to this idea that, as all physical things are instantiations of an ideal in the sensible world, the existence of vacuum would need an ideal form of vacuum, which he cannot conceive as reasonable. As for him, Aristotle considers the vacuum as impossible to be created, arguing that nothing cannot be obtained from something.
In medieval Europe the concept of vacuum itself was considered as heretic, as it implies the absence of God [7]. In the beginning of the 17th century, Galileo nevertheless introduced the vacuum in its mechanical laws, as the limit of a non-frictious medium, when studying the acceleration of falling objects [8]. Descartes [9, II; 16] in 1644 rejects the theory of vacuum: as he refuses the distinction between object and space, and states that the existence of the object implies the existence of space, he deduces the impossibility of vacuum. The same year Toricelli [10] remarked that mercury only goes up by 760 mm when pumped in a tube, a phenomenon that Pascal [11, 12] explained a few years later by the existence of an atmospheric pressure. Against the common idea that « matter hates vacuum », he shows that vacuum appears in the upper space of the tube when the liquid is pumped.
In 1801, while Newtonian physics had successfully described light as tiny particles moving in vacuum for years, the appearance of interference effects in Young’s diffraction experiment [13] gives evidence of the wave nature of light, already predicted by Hooke [14] and Huygens [15] in the 17th century, and foreseen by Euler [16] in the 18th century. Sound is a vibration of the air, and light propagates in the vacuum, the scientist therefore assume that light must have a proper vibrating media, which they call the luminiferous aether [17].
In 1887, the Michelson-Morley experiment [18, 19] aiming at measuring the speed of the aether with respect to the earth turned out to be inconclusive. The solution came from Einstein special relativity [20], which disproves the existence of an aether. The light is then described as a wave of the electromagnetic field (E; B), travelling in vacuum at an equal velocity c for any direction, and in any unaccelerated frame, following the Maxwell equations [21]. Finally, the classical vacuum can be defined as space when all matter has been removed and when the electromagnetic field is zero: Classical vacuum: E = B = 0 :

The quantum vacuum

The advent of Quantum Theory has deeply changed our understanding of many basic physical concepts such as light, matter, and movement. One example of a new idea brought by Quantum Theory is the Heisenberg uncertainty2 principle [22], stating the impossibility of measuring the position x and the momentum px of a particle with an unlimited accuracy: x px 2 where x and px are the indetermination of x and px, respectively, and ~ is the reduced Planck’s constant. For the simple case of a harmonic oscillator model, such as a mass attached to a spring, this leads to the impossibility for the objet to stand still in a lower position with no oscillations. The configuration with the minimum of movement is then a tiny rocking, associated to a non-zero energy E0 > 0.
In Quantum Field Theory, each mode of frequency ! of the electromagnetic field can be described as a harmonic oscillators [26, 27, 28]. The Heisenberg uncertainty principle can then be applied, and yields that the electromagnetic field must undergo unavoidable fluctuations around zero, and that the electromagnetic energy cannot decrease below a zero-point energy E0 = ~2! . In opposition to the classic vacuum, the electromagnetic fields are not zero anymore, but rather fluctuate around
0: Quantum vacuum: E = B = 0 but E2; B2 > 0 where E is the average of E over time. Quantum Theory has changed dramatically our conception of vacuum, as an active and dynamic media, undergoing fluctuations of the electromagnetic field and creation of particle pairs. The discontinuity between matter (or light) and vacuum has been removed, and philosophically speaking, the concept of vacuum now lies much closer to the being than to the nothingness.

The Casimir effect

The van der Waals force [29] describes the interaction between neutral atoms or molecules, and plays a crucial role in colloid chemistry, nanotechnologies and surface science. An important contribution to this interaction is done by the London dispersion forces [30, 31, 32, 33], arising from quantum induced instantaneous polarization multipoles in molecules.
Studying the stability of colloids, Verwey and Overbeek observed [34] that the van der Waals-London interaction seems to decrease at long range more rapidly than the L 6 predicted power-law. Overbeek pointed out that forces need time to propagate, while van der Waals-London forces were derived by considering instantaneous interactions. In 1946 Casimir and Polder manage to include the effects of retardation [35], which becomes non-negligible for large distances and change the power-law in the distance from L 6 to L 7. In their work they also include the situation of an atom facing a perfectly reflecting surface, taken « as a preliminary exercice » [36], bearing the famous Casimir-Polder potential: 3~c ECP= 8 L4 where is the atomic polarizability. The obtained formula are simple and elegant and Casimir was sure that there should be an elegant way to derive them. Advised by Bohr [36], Casimir interpreted the retarded van der Waals-London interaction as an effect of the quantum vacuum zero-point energy [37]. He also realised in 1948 that his founding could be applied to the case of two perfectly reflecting parallel mirrors [38], and obtained the energy per unit area: ECas ~c 2 = ; A 3 720L giving birth to the famous Casimir effect, an observable manifestation of quantum vacuum fluctua-tions in the macroscopic world [39]. In this formula, the mixture of Maxwell’s electromagnetism and Quantum Theory in the Casimir effect is clearly illustrated by the two terms c and ~.

Applications

The Casimir force is the most accessible experimental effect of vacuum fluctuations in the macro-scopic world. As the non-zero vacuum energy is known to raise serious difficulties at the interface of quantum theory and gravitational effects, it is worth investigating the Casimir effect with the greatest care and highest accuracy, in order to test the predictions of Quantum Field Theory [40, 41, 42, 43, 44, 45, 46, 47, 48, 49] and its connection with the problem of zero-point energy [50, 44, 51, 52]. Moreover, in various accurate force measurements in the nanometer to micrometer range, the Casimir effect becomes the dominant force and a precise knowledge of its magnitude is often a key point. The most obvious case are the tests of Newtonian gravity [53, 54, 55, 56, 57, 58, 59, 60, 61, 62], and the search for new weak forces predicted in theoretical unification models in the nanometric to millimetric range [63, 64, 65, 66, 67, 68, 69, 70]. Assuming a corrective term to the Newtonian gravity in the Yukawa form: U(L) = G m1m2 1 + e L those studies aim at giving constraints on the possible range and relative amplitude . Various other experiments have their results « contaminated » by the Casimir effect. For molecular interferometry [71, 72, 73, 74, 75, 76], because of the increased size and the reduced speed of the molecules, the Casimir attraction to the grating deviates the molecules when it passes through the grating, leading to an effective smaller slit width. In some cases, the Casimir interaction is too strong [77, 78] and has to be avoided, for instance with the use of light gratings [79]. In Section 12 (p.158) we will mention the study the intriguing phenomenon of small heating of ultracold neutrons (UCNs) in traps [80, 81, 82], for which an accurate knowledge of the quantum states of nanospheres interacting with surfaces is needed. As we will see, this has to be done through a careful treatment of the Casimir potential undergone by the nanospheres.
Finally, because the Casimir and van der Waals forces are dominant at the scale of the micrometer and below, they enter various important domains such as atomic and molecular physics, condensed matter and surface physics, chemical and biological physics, and micro and nanotechnologies [83]. For the latter they are important in the architecture of micro and nano-oscillators (NEMS and MEMS), not only to avoid stiction [84, 85, 86, 87], but also to be used as an actuator [49, chap.8].

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Experimental works

The first experiment to measure qualitatively the Casimir force is performed in 1958 by Sparnaay [88], using a spring balance to measure the force between two flat neutral metallic plates. The measurements were carried out for distances between 0.5 and 2 m, and the poor experimental accuracy led the author to the conclusion that « the observed attractions do not contradict Casimir’s theoretical prediction ».
A major difficulty in the former experiment being to keep a good parallelism between the two plates, a first unambiguous measurement of the Casimir force could only be achieved twenty years later by van Blockland and Overbeek [89], by using a lens and a flat plate covered with chromium layers. The measurements were carried out for distances between 132 to 760 nm and led to an agreement with the theory that can be estimated around 25% [55]. Other early experimental efforts have measured the Casimir force in those years, such as [90, 91, 92, 93], the reader is referred to detailed and systematic reviews that may be found in [94, 40, 95, 41, 42, 55].
In the past years, a series of new measurements with improved accuracy have been possible thanks to new techniques. In 1997 Lamoreaux measured the Casimir force between a metallized sphere and a flat metallic plate, using a torsion pendulum [96, 97, 98, 99, 100] for distances between 0.6 and 6 m with an agreement to theory that can be evaluated to 10% for the shortest distances [55]. For the largest distances, the weak magnitude of the force prevented the observation of temperature corrections.
A second precise measurement of the Casimir effect has been done in 1998 by Mohideen [101, 102, 103]. The experimental setup, presented in the left part of Fig. 13, consists on a metallized sphere attached to the cantilever of an atomic force microscope (AFM), brought close to a metallic plate. The Casimir attraction between the sphere and the flat plate yields a bent of the cantilever, which is measured by the deflection of a laser beam on the top of the cantilever. The distance range, from 0.1 to 0.9 m, is smaller than in the previous experiment, enabling an experimental accuracy at the level of 1%.
In 2001 Capasso and his group measured the Casimir force gradient using a microelectromechan-ical system (MEMS) [85, 104] for distances between 0.1 and 1 m. The presence of a polystyrene sphere with metallic coating close to a polysilicon plate, also with metallic coating, changes its oscillation frequency in agreement with the prediction from the Casimir effect.
With a similar technique, presented in the right part of Fig. 13, Decca carried out in 2003 one of the most precise experiment on the Casimir effect [105, 46, 106, 107, 60, 108], for distances between 0.2 and 2 m. The experimental accuracy, below the percent level for small distances, enable to observe a discrepancy with the theory that the author attribute to a poor characterisation of the optical properties of the chosen materials. Moreover, the experimental data surprisingly seem to favour a description of metals by the loss-less plasma model, with respect to a description with the dissipative Drude model [107, 109]: the experimental points are closer to theoretical curves derived with the plasma model (see Fig. 14), and no thermal effect is observed at short distances as it should with the Drude model.
Very recently, Lamoreaux conducted another experiment [110, 111] with a torsion pendulum attached to a very large spherical lens, in order to measure the Casimir force for larger distances from 0.7 to 7 m. The conclusions are opposite to the ones in [107], as the author reported that « the experimental results are in excellent agreement with the Casimir force calculated using the Drude model » and that « plasma model result is excluded in the measured separation range ». Moreover, thermal effect in the Casimir force is clearly observed, thanks to the larger distance range.
With this review of experiments dedicated to the Casimir effect, we observe that the sphere-plane geometry is the configuration of the most precise measurements, and therefore the important one in the prospect of gravity tests. Some other configurations have been studied, such as non-parallel cylinders [112], corrugated plates [113, 114], or even the original Casimir configuration of two flat plates [115, 116, 117], for which an experimental accuracy of the order of 15% could be achieved for distances between 0.5 and 3 m. This remark stresses the importance of the theoretical investigation of the sphere-plane configuration for accurate comparison between theory and experiments.

Table of contents :

Introduction 
I. Reminder on the use of the scattering formalism for Casimir computations, and example of the plane-plane geometry 
1. The Scattering formula
1.1. The planar electromagnetic modes
1.2. The cavity operator
1.3. The Casimir energy from the radiation pressure
1.4. The example of the plane-plane geometry
2. Optical properties of materials
2.1. The dielectric function
2.2. The plasma model
2.3. The Drude model
2.4. Dielectrics
2.5. The example of the plane-plane geometry
3. The scattering formula at non-zero temperature
3.1. The Matsubara sum
3.2. The example of the plane-plane geometry
4. The proximity force approximation (PFA) 55
4.1. The PFA formula in the sphere-plane geometry
4.2. The primo-potential D
4.3. Beyond the PFA method
II. Method for the Casimir effect in the sphere-plane geometry 
5. The scattering formula in the sphere-plane geometry
5.1. Derivation of the scattering operator D({)
5.2. Explicit form of the various involved quantities
5.3. Transformation to real and simpler quantities
5.4. Conclusion
6. Numerical issues
6.1. From operators to matrices: the truncation to `max
6.2. Differentiations with respect to L and T
6.3. Integration/sum over ~
6.4. Integration over cos
6.5. Modified Bessel functions
6.6. Numerical stability
6.7. Computation of the zero-frequency term
III. Analytical limits 
7. Low-frequency limit
7.1. Mie coefficients
7.2. Fresnel coefficients
7.3. Spherical harmonics and finite rotations
7.4. Integration over cos
7.5. Determinant of the scattering matrix
7.6. Conclusion: the first Matsubara term and the high-temperature limit
8. Long-distance limit
8.1. The dipolar-simple scattering approximation
8.2. Perfect mirrors
8.3. Metallic scatterers modelled with the plasma model
8.4. Metallic scatterers modelled with the Drude model
8.5. Dielectric scatterers
IV. Results at zero temperature 
9. Beyond-PFA computations in the literature
9.1. Ways to measure the accuracy of PFA
9.2. Scalar results for perfect mirrors
9.3. Electromagnetic results for perfect mirrors
9.4. Experimental prescription on and conclusion
10.Results for perfect mirrors
10.1. Behaviour of numerical results at short and large separation
10.2. Power laws
10.3. Beyond-PFA corrections
11.Results for metallic mirrors
11.1. Observation of the effect of imperfect reflection
11.2. Power laws
11.3. Correlations between the effects of finite conductivity and geometry
11.4. Influence of conductivity on the beyond-PFA corrections
12.Results for dielectric nanospheres
12.1. The Casimir-Polder formula for a dielectric nanosphere
12.2. The complete multipolar expression E
12.3. Averaging Casimir-Polder over the sphere’s volume
V. Results at non-zero temperature 
13.Perfect mirrors at ambient temperature
13.1. Observation of the thermal effects
13.2. Power laws
13.3. Correlations between curvature and thermal effects
13.4. Casimir entropy
13.5. Comparison with PFA at short distance
14.Metallic mirrors at ambient temperature
14.1. Influence of temperature for metallic materials
14.2. Influence of imperfect reflection at ambient temperature
14.3. Study of various interplays
14.4. Beyond-PFA corrections
14.5. Power laws
15.High-temperature regime
15.1. Perfect mirrors
15.2. Drude model for metallic mirrors
15.3. Ratio of perfect mirrors result over Drude metals result
Conclusion and outlook 
Appendix 
A. Proofs of lemmas, properties and theorems
A.1. Lemma 1 (p.48)
A.2. Lemma 2 (p.48)
A.3. Property 4 (p.48)
A.4. Property 5 (p.75)
A.5. Property 6 (p.86)
A.6. Property 12 (p.101)
A.7. Property 13 (p.114)
A.8. Property 14 (p.128)
B. Approximations methods for the sphere-plane geometry
B.1. PFA methods
B.2. PWS methods
Published articles
Casimir Interaction between Plane and Spherical Metallic Surfaces (PRL, 2009)
Thermal Casimir Effect in the Plane-Sphere Geometry (PRL, 2010)
Thermal Casimir effect for Drude metals in the plane-sphere geometry (PRA, 2010)
Casimir interaction between a dielectric nanosphere and a metallic plane (PRA, 2011)
References 

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