The present notes are organized as the lectures given at the Les Houches Summer School “Quantum Optics and Nanophotonics” in August 2013. The first section contains an introduction and a description of the current state-of-the-art for Casimir force measurements and their comparison with theory. The second and third sections are a pedagogical presentation of the main features of the theory of Casimir forces for 1-dimensional model systems and for mirrors in 3-dimensional space.
The emergence of quantum theory has profoundly altered our conception of space by forcing us to consider it as permanently filled by vacuum field fluctuations Milonni1994 (); Milton2001 (). These vacuum fluctuations are electromagnetic fields propagating with the speed of light, as any free field, and corresponding to an energy of half a photon per mode. They have a number of observable consequences in microscopic physics, for example the radiative corrections in subatomic physics Itzykson1985 (), the spontaneous emission processes, the Casimir-Polder interaction and Lamb shift in atomic physics Cohen-Tannoudji1992 ().
Vacuum fluctuations also have observable mechanical effects in macroscopic physics and the archetype of these effects is the Casimir force between two motionless mirrors. This force was predicted in 1948 by H. B. G. Casimir Casimir1948 () and soon observed in experiments Lamoreaux1999 (). Experiments have been improved after years of development and they now reach a good level of precision Decca2003 (); Decca2005ap (); Decca2007prd (); Decca2007epj () (more references below). However, the comparison with theoretical predictions has raised difficulties which have been discussed in a large number of papers (references in Klimchitskaya2009 (); Dalvit2011 ()).
This comparison is particularly interesting because of its fascinating interfaces with open questions in fundamental physics. The Casimir effect is connected with the puzzles of gravitational physics through the problem of vacuum energy as well as with the principle of relativity of motion Jaekel1997 (). Effects beyond the Proximity Force Approximation also make apparent the rich interplay of vacuum and geometry Balian2003 (). Casimir physics also plays an important role in the tests of gravity at sub-millimeter ranges Fischbach1998 (); Adelberger2003 (); Adelberger2009 (); Antoniadis2011 (). For scales of the order of the micrometer, such gravity tests are performed by comparing Casimir force measurements with theory and the comparison has to take into account the many differences between real experiments and the idealized case considered initially by Casimir Lambrecht2003 ().
At the end of this short general introduction, it has also to be stressed that the Casimir and closely related Casimir-Polder forces Casimir1948pr (); Feinberg1970 (); Power1993 (); Milton2011ajp (); Intravaia2011 () have strong connections with various active domains and interfaces of physics, such as atomic and molecular physics, condensed matter and surface physics, chemical and biological physics, micro- and nano-technology Parsegian2006 (); French2010 ().
The present paper is organized as the lectures given at the Les Houches Summer School in August 2013 : the first section contains an introduction and a description of the current state-of-the-art for experiments and their comparison with theory while the second and third sections provide a pedagogical presentation of the main features of the theory of Casimir forces.
Section I begins with a short history of quantum field fluctuations in vacuum. We then review various arguments involved in the comparison with theory of the experiments devoted to the measurement of the Casimir force. As experiments are performed with gold-covered plates, the force depends on non universal properties of the real plates used in the experiments. As they are performed at room temperature, the effect of thermal field fluctuations has to be added to that of vacuum fluctuations. The most precise experiments are performed in the plane-sphere geometry and not in the geometry of two parallel planes whereas the latter is theoretically easier to handle. Finally, surfaces are non ideal, and effects such as roughness, electrostatic patches and contamination affect the comparison between theory and experiment.
Section II contains a simple derivation of the Casimir effect in a model of scalar fields on a 1-dimensional line. This model allows one to introduce the Quantum Optics approach to the Casimir effect. This approach is based on the existence of field fluctuations which pervade empty space and exert radiation pressure on mirrors at rest in vacuum. The force is thus calculated as the result of different pressures acting on inner and outer sides of the two mirrors which form a cavity. This approach is often also called the Scattering Formalism because all properties of the Casimir force are determined by the reflection amplitudes of the fields on the two mirrors Jaekel1991 ().
Section III then treats the case of two plane and parallel mirrors at rest in electromagnetic vacuum in 3-dimensional space. It describes the models generally used for the metallic mirrors used in the experiments Lambrecht2000 (); Genet2003 () and discusses the results obtained in this manner. For mirrors characterized by Fresnel reflection amplitudes deduced from a linear and local dielectric function, the Scattering Formalism leads to the same results as Lifshitz’s method Lifshitz1956 (); Dzyaloshinskii1961 (); Lifshitz1980 (). The section ends up with a presentation of the general scattering formalism which allows one to deal with non specular reflection and arbitrary geometries Lambrecht2006 (); Emig2007 (); Milton2008 (); Lambrecht2011 (); Rahi2011 (); Rodriguez2011 ()
I Vacuum fluctuations and Casimir forces
The birth of quantum vacuum
The classical idealization of absolute empty space was affected by the discovery of black body radiation which is present everywhere at non zero temperature and exerts a pressure onto the boundaries of any cavity. It is precisely for explaining the properties of black body radiation that Planck introduced the first quantum law in 1900 Planck1900 () (discussions in Darrigol2000 ()).
In modern terms, the first Planck law gives the energy per electromagnetic mode characterized by its frequency as the product of the energy of a single photon by a mean number of photons per mode
Unsatisfied with his first derivation, Planck Planck1911 () wrote in 1911 a new expression for the mean energy per mode
The difference between the two Planck laws corresponds to the zero-point energy. Whereas the first law describes a cavity entirely emptied out of radiation at the limit of zero temperature ( when ), the zero-point energy added in the second law persists at zero temperature.
The story of the two Planck laws and of the discussions they raised is related for example in Milonni1991 (). The arguments used by Planck in 1911 are no longer considered as satisfactory nowadays but an argument which is still valid was proposed by Einstein and Stern in 1913 Einstein1913 (). In order to state this argument, let us consider the limit for at high temperatures ()
In contrast to the first Planck’s law (1) which falls off the correct classical limit by a constant offset , the second Planck’s law (2) matches the correct classical limit at high temperatures. This feature is emphasized by the modern writing of this law which has clearly no term linear in because the right-hand side is an even function of (note that )
Debye was the first to insist on observable consequences of zero-point fluctuations in atomic motion, by discussing their effect on the intensities of diffraction peaks Debye1914 () whereas Mulliken gave the first experimental proof of these consequences by studying vibrational spectra of molecules Mulliken1924 (). At this point, we may emphasize that these discussions took place before the existence of these fluctuations was confirmed by fully consistent quantum theoretical calculations. Nowadays, vacuum fluctuations are just an immediate consequence of Heisenberg inequalities (see for example references in Reynaud1990 (); Reynaud1992 (); Reynaud1997 ()).
The puzzle of vacuum energy
Nernst is credited for having been the first to emphasize in 1916 that zero-point fluctuations should also exist for free electromagnetic fields Nernst1916 () (discussions in Browne1995 ()), thus discovering what physicists now call quantum vacuum. He also noticed that the associated energy constituted a challenge for gravitation theory. When summing up the zero-point energies over all field modes, a finite energy density is obtained for the first Planck law - this is the solution of the ultraviolet catastrophe - but an infinite value is produced from the second law. When introducing a high frequency cutoff, the calculated energy density remains finite but it is still much larger than the mean energy observed in the world around us through gravitational phenomena Weinberg1989 (). Pedagogical derivations and numbers illustrating this major problem are given in Adler1995 ().
This puzzle has led famous physicists to deny the reality of vacuum fluctuations. In particular, Pauli stated in his textbook on Wave Mechanics Pauli1933 () (translation from Enz1974 ()): At this point it should be noted that it is more consistent here, in contrast to the material oscillator, not to introduce a zero-point energy of per degree of freedom. For, on the one hand, the latter would give rise to an infinitely large energy per unit volume due to the infinite number of degrees of freedom, on the other hand, it would be principally unobservable since nor can it be emitted, absorbed or scattered and hence, cannot be contained within walls and, as is evident from experience, neither does it produce any gravitational field.
A part of these statements is simply unescapable: it is just a matter of evidence that the mean value of vacuum energy as predicted by quantum theory does not contribute to gravitation as an ordinary energy. However, we know nowadays that vacuum fluctuations are scattered by matter, as shown by the numerous effects of the associated scattering in subatomic Itzykson1985 () and atomic Cohen-Tannoudji1992 () physics. And the Casimir effect discussed in the sequel of this paper may be seen as the manifestation of vacuum fluctuations when being contained within walls. Let us note at this point that different points of view coexist about the significance of vacuum fluctuations Enz1974 (); Schwinger1975 (); Aitchinson1985 (); Saunders1991 (); Sciama1991 (); Rugh2002 (); Jaffe2005 (); Kragh2012ahes (); Kragh2012aag ().
The puzzle of vacuum energy has been discovered nearly one century ago and it is not yet solved. It has led and still leads to many ideas: for example, a dark energy length scale =85m can be defined by setting the cutoff used to calculate the vacuum energy so that it fits the now measured cosmic vacuum energy density Riess1998 (); Perlmutter1999 (); Bennett2003 (). It is thus natural to check if gravity could be affected below this dark energy length scale Beane1997 (); Dvali2001 (); Sundrum2004 (). It has been shown in torsion-balance experiments Kapner2007 () that Yukawa modifications of the gravitational inverse-square law can have a magnitude equal to that of gravity only if their range has a value smaller than 56m, which discards this kind of ideas under their simplest forms. More possibilities, corresponding for example to power-law modifications associated with compact extra-dimensions Arkani-Hamed1998 (), are discussed in Adelberger2003 (); Adelberger2009 (); Antoniadis2011 ().
The search for scale-dependent modifications of the gravity force law are currently pushed down to even smaller ranges and they approach the micrometer distance range where Casimir forces are predominant. For these Casimir tests of the gravity law to make sense, the accuracy and reliability of theoretical and experimental values have to be assessed cautiously and independently of each other. In particular, systematic effects have to be identified and eliminated, whenever this is possible. This implies in particular to deal carefully with the many differences between the idealized situation studied by Casimir and the configuration of real experiments Lambrecht2012 ().
The Casimir force between ideal and real mirrors
In his initial calculation Casimir1948 (), Casimir considered an idealized configuration, with perfectly smooth, flat and parallel plates (see Figure 1) in the limit of zero temperature and perfect reflection. denotes the distance between the two plates and their area, supposed to be large enough ().
He thus obtained expressions for the energy and force which exhibit a universal behavior associated with the confinement of vacuum energy
where is the speed of light and the reduced Planck constant. The signs have been chosen according to the standard thermodynamical conventions (the relation with thermodynamics of the Casimir effect will be discussed later on). The negative energy corresponds to a binding energy and the negative force to an attractive force, that is also a negative pressure
The order of magnitude of the pressure is 1mPa for two mirrors at the distance m typical for Casimir force measurements (see below). The formula (6) describes an extremely rapid increase of the pressure when the distance is decreased, and it would lead to a value 1TPa typical of strong molecular cohesion when it is extrapolated down to atomic distances 0.1nm. This means that the Casimir force is a quantum force like molecular cohesion forces, which has a weaker magnitude only because it is measured at distances much larger than typical atomic distances. Note however that formula (6) cannot be used at atomic distances where the ideal assumptions used to derive it are no longer valid, as we explain now.
The effect of imperfect reflection
Indeed, perfectly reflecting mirrors do not exist, except as idealizations giving fair descriptions of reality in limiting cases only. The mirrors used in the experiments are made of metal and they have a good reflection only at frequencies below the plasma frequency. Accounting for their imperfect reflection and its frequency dependence is thus essential for obtaining a reliable theoretical expectation of the Casimir pressure Lambrecht2003 (). In other words, the real Casimir pressure depends on the non universal optical properties of the material plates used in the experiments. It can be written as the product of the ideal result (6) by a dimensionless factor which accounts for these optical properties
The expression of in terms of the optical properties of the mirrors will be given in section III.
Most descriptions of the metallic mirrors used in the experiments are based on Fresnel reflection laws at the two interfaces between vacuum and metallic bulks with optical properties described by a linear and local dielectric response function. This dielectric function is a sum of contributions corresponding respectively to bound () and conduction electrons, the latter being directly related to the conductivity ()
Note that functions , and are all defined as reduced quantities, with their SI counterparts being , and ( is the vacuum permittivity). With these conventions, and are dimensionless while has the dimension of a frequency.
The dielectric function (8) has to be obtained from optical data Lambrecht2000 (); Svetovoy2008prb () as they are tabulated for example for gold in Palik1995 (). At low frequencies, tends to a constant while the contribution of conduction electrons diverges while tends to a constant . Optical data have then to be extrapolated at low frequencies by using the dissipative Drude model for the conductivity of the metal Ashcroft1976 ()
Here is the plasma frequency and the relaxation parameter for conduction electrons. This model meets the well-known fact that gold has a finite static conductivity
For reasons which will become clear in the following, the limiting case of a lossless plasma of conduction electrons ( in (9)) is often considered
This so-called plasma model cannot be an accurate description of metallic mirrors. As a matter of fact, it contradicts the fact that gold has a finite static conductivity (10) while also leading to a poorer fit of tabulated optical data than the more general Drude model Lambrecht2000 (). However, is much smaller than for a good metal (for example for gold). As the difference between (9) and (11) is appreciable only at low frequencies where is very large for both models, one might expect that it does not affect too much the value of the Casimir force. This naive expectation is met at small distances or low temperatures but not in the general case of arbitrary distances and temperatures, as explained in the following.
The effect of temperature
Most experiments are performed at room temperature, so that the effect of thermal fluctuations has to be added to that of vacuum fields Mehra1967 (); Brown1969 (); Schwinger1978 (); Genet2000 (). This important point will be discussed in a detailed manner in sections II & III. At this point, we focus on the strong correlation effect obtained between the effects of temperature and dissipation, which has given rise to a large number of contradictory papers (see for example the references in Milton2005 (); Klimchitskaya2006 (); Brevik2006 (); Ingold2009 (); Brevik2014 ()).
Boström and Sernelius Bostrom2000 () were the first to remark that, in spite of the naive expectation described in the end of the preceding subsection, dissipation has a large effect on the value of the Casimir force at distances accessible in experiments and at room temperature. Their result is illustrated on Figure 2 where the ratio giving the real Casimir pressure (7) is drawn for the Drude and plasma models at room temperature (=300K), using the formulas in Ingold2009 (). These two models correspond to the dielectric function (8) with =1 and substituted by (9) and (11) respectively. The simplification =1 does not change the difference between the predictions of the two models and it can also be dropped, by having deduced from the optical data. The parameters of the optical models are chosen to match values typical for thick layers of gold ( with =136nm and ).
A striking difference appears between the predictions of the two models Bostrom2000 (). These predictions, which are close to each other at short distances, exhibit an increasing difference for distances of the order or larger than 1m. In particular, the ratio of the plasma to Drude prediction for the Casimir pressure goes to a factor 2 at the limit of large distances. In fact, the result of the plasma model coincides at this limit with that obtained for perfect mirrors whereas the result of the Drude model reaches only half that value. It is worth recalling here that this last result is reproduced by the derivations of Casimir pressures from microscopic models of the metallic mirrors Jancovici2005 (); Buenzli2005 (); Bimonte2009 ().
As a matter of principle, there should be no doubt that the Drude model is a better representation of the optical properties of real plates at low frequencies than the plasma model. At this point however, we have to face discrepancies in the comparison between experimental results and theoretical predictions and, unexpectedly, some experimental results appear to lie closer to the predictions of the plasma model than to that of the Drude model Klimchitskaya2009 (). Before coming to this point, we have still to discuss another important feature of the recent precise experiments which are performed in the plane-sphere geometry and not in the plane-plane geometry in which most calculations are done.
The effect of geometry
The configuration used for most Casimir experiments corresponds to a plane and a sphere, with the distance of closest approach and the radius of the sphere, supposed to be large (see Figure 3). The force in this plane-sphere geometry is usually calculated by using the so-called Proximity Force Approximation (PFA) Derjaguin1956 (); Decca2009 (). Let us emphasize here that this approximation is valid only at the limit of a very large radius () and that the question of its accuracy for a finite value of remains an open question (more discussions on this topic later on).
We assume here provisionally that the PFA is precise enough for the purpose of theory-experiment comparison. It follows that the force between the plane and the sphere can be obtained by integrating over the distribution of local inter-plate distances the pressure calculated in the geometry with two parallel planes. In the plane-sphere geometry, this gives
where is the pressure evaluated between two planes at at a distance from each other; runs over distances larger than the distance of closest approach and is the corresponding element of surface.
In the experiment, the gradient of the Casimir force in the plane-sphere geometry is measured (see the next subsection). This quantity is deduced from (12) as
We now present briefly the experimental methods and results. To this aim, we focus our attention on a few experiments : the experiment at IUPUI Decca2005ap (); Decca2007prd (); Decca2007epj () which has been run for ten years, with results pointing to an unexpected conclusion later confirmed at UCR Chang2012 (), and the experiment in Yale Sushkov2011 () which points to a different conclusion. We also give here a list of other Casimir measurements which have produced information of interest on the topics discussed in this paper Derjaguin1956 (); Sparnaay1958 (); Sabisky1973 (); Lamoreaux1997 (); Mohideen1998 (); Harris2000 (); Ederth2000 (); Chan2001 (); Bressi2002 (); Lisanti2005 (); Decca2005prl (); Kim2008 (); Jourdan2009 (); Man2009 (); Masuda2009 (); Munday2009 (); Torricelli2010 (); Banishev2012 (); Banishev2013 (); Castillo-Garza2013 ().
The experiment at IUPUI is described in the papers Decca2005ap (); Decca2007prd (); Decca2007epj (). A summary and update can be found in the slides associated with a talk given recently by R.S. Decca at a Pan-American Advanced Study Institute school Decca2012pasi (). The experiment uses dynamic measurements of the resonance frequency of a torsion micro-oscillator. For the free micro-oscillator, that is in the absence of the Casimir force, the resonance frequency is determined by the stiffness coefficient and the moment of inertia
When a gold-covered sphere is approached from the gold-covered plane of the micro-oscillator plate, the effective stiffness is modified as the gradient of the Casimir force . The resonance frequency is thus shifted to a new value
where is the lever arm. As the radius of the sphere is m and the range of distances mm, the condition is met. Using the expression (13) which gives the gradient within the PFA, and measuring accurately , and , the shift of the squared frequency is then transformed into a reading of the Casimir pressure as it would be between two planes at distance
is given by (7) and will be discussed in section III. Note that the separation between bodies is measured separately through two-color interferometry, up to a global offset adjusted in the data analysis process Decca2007prd (); Decca2007epj ().
When compared with the theoretical prediction, this measurement leads to unexpected conclusions Decca2007prd (); Decca2007epj (): the measurements appear to agree with the predictions obtained from the lossless plasma model but to deviate significantly from those deduced from the Drude model which accounts for dissipation (see Fig.1 in Decca2007prd ()). These experiments are performed in a range of distances mm where the difference between the predictions of the two models is small. This entails that the problem of accuracy, that is also the control of systematic errors, is a critical issue. However, the deviation of experimental results from theoretical expectations (based on the Drude model) is clearly larger than the statistical dispersion of these results (bars on Fig.1 in Decca2007prd ()). More details on statistical and systematic errors in this experiment can be found in Decca2007prd (); Decca2007epj ().
Different conclusions are reached in a more recent experiment performed in Yale Sushkov2011 (). This experiment aims at measurements at larger distances mm, where the force is smaller while the thermal contribution and the effect of dissipation are larger (see Fig.2). The experimental technique is based on a torsion balance and uses a much larger sphere mm, which allows for measurements of weaker pressures. This experiment clearly sees the thermal effect and its results fit the predictions drawn from the dissipative Drude model, after the contribution of the electrostatic patch effect has been subtracted Sushkov2011 (). These new results have to be confirmed by further studies Milton2011natphys (). The main issue in this experiment is that the pressure due to electrostatic patches is larger than that due to Casimir effect, so that a proper modeling of this contribution is critical whereas the patch pattern has not been characterized independently. This is in fact a more general problem since the patch properties have not been measured in other experiments either (more discussions below).
The conclusion at this point is that the Casimir effect is measured with a good precision in several experiments, with a persisting problem however in terms of accuracy. The results of the most precise experiment, improved over a decade at IUPUI and confirmed recently at UCR, appear to favor theoretical predictions obtained with the lossless plasma model and to deviate from the predictions obtained with the best motivated model, that is the dissipative Drude model. The Yale experiment fits predictions drawn from this Drude model, after the subtraction of a large contribution of the electrostatic patch effect. For the IUPUI experiment, the pressure difference goes up to 50mPa at the smallest distances 160nm where the pressure itself is 1000mPa, which entails that the accuracy is certainly not at the 1% level, as has been occasionally claimed.
The difference between the experimentally measured () and theoretically predicted () values of the Casimir pressure is drawn on Fig.4 as a function of the distance . Experimental values and error bars correspond to data kindly provided by R.S. Decca Decca2007prd (); Decca2007epj (); Decca2012pasi (). Theoretical values were calculated by R.O. Behunin et al in Behunin2012pra85 (), using the optical data of gold extrapolated at low frequencies to a Drude model, and at room temperature. Systematic corrections were done in Behunin2012pra85 () as in Decca2007prd (); Decca2007epj () and similar results obtained. The discrepancy clearly appears on Fig.4 and it is of particular importance in the context of gravity tests at sub-millimeter ranges Adelberger2009 (); Antoniadis2011 (). The deviation seen on Fig.4 does not look like a Yukawa law, but it certainly looks like a combination of power laws !
This discrepancy between theory and experiment may have various origins, in particular artifacts in the experiments or inaccuracies in the theoretical evaluations. They may also come from yet unmastered systematic effects in the comparison between experimental data and theoretical predictions. They could in principle be the first hint of the existence of new forces beyond the standard model, though such a strong statement should only be considered after a cautious examination of the more mundane explanations associated in particular with systematic effects.
The theoretical formula used to calculate the Casimir pressure between real plates will be derived in section III. It will reproduce the ideal Casimir expression at the limits of perfect reflection and null temperature while being valid at any temperature for any model of mirrors obeying well motivated physical properties Jaekel1991 (); Lambrecht2000 (), including the case of dissipative mirrors Genet2003 (). When the reflection amplitudes are deduced from the Fresnel laws, and semi-infinite bulk mirrors are characterized by a linear and local dielectric response function, the results reproduce those of I.E. Dzyaloshinskii, E.M. Lifshitz and L.P. Pitaevskii Lifshitz1956 (); Dzyaloshinskii1961 (); Lifshitz1980 (). It then remains to specify this dielectric function and its low-frequency behavior. Here, it is worth emphasizing that the Drude model, though being obviously much better motivated than the lossless plasma model, is not a very accurate description of conduction phenomena in real metals. More detailed descriptions can be considered, which can for example be determined from microscopic models of conduction in metals. Attempts in this direction and discussions can be found for example in Pitaevskii2008prl (); Dalvit2008prl (); Svetovoy2008prl (); Geyer2009prl (); Pitaevskii2009prl (); Decca2009prl (); Dalvit2009prl (). To date, they have been unable to explain the discrepancy.
A possible source of systematic error is the use of the Proximity Force Approximation (PFA) in order to derive expressions for the plane-sphere geometry from those known from the plane-plane geometry. This approximation is expected to be valid at the limit where the aspect ratio goes to zero. Even in this case, the accuracy of the PFA for a finite value of remains an open question after the remarkable advances made recently on this topic Canaguier-Durand2012ijmpcs (), which will be described in section III. The question remains open, even though most specialists would probably bet that the deviation from PFA is not able to bridge the gap between experiment and theory. Other possible sources of systematic error involve the effects of surface physics on Casimir experiments. The problem of surface roughness has been studied in a thorough manner Neto2005epl (); Neto2005pra (); Zwol2009 (); Broer2011 (); Broer2012 (); Broer2013 ().
Electrostatic patches and contamination
Electrostatic patches and contamination, already alluded to, are a worrying source of such systematic effects, discussed in the sequel of this section. Electrostatic patches have been known for a long time to be a source of worries for a large number of high precision measurements Witteborn1967 (); Camp1991 (); Sandoghdar1996 (); Turchette2000 (); Deslauriers2006 (); Robertson2006 (); Epstein2007 (); Pollack2008 (); Dubessy2009 (); Carter2011 (); Everitt2011 (); Reasenberg2011 (); Hite2012 (); Hite2013 (), and in particular for Casimir experiments Speake2003 (); Chumak2004 (); Kim2010pra (); Kim2010jvst (); Man2010 () and short-range gravity tests Adelberger2009 (). The patch effect is due to the fact that the surface of a metallic plate cannot be an equipotential as it is made of micro-crystallites with different work functions. For clean metallic surfaces studied by the techniques of surface physics, the resulting voltage roughness is correlated to the topography roughness as well as to the orientation of micro-crystallites Gaillard2006 (). For surfaces exposed to air, the situation is changed due to the unavoidable contamination by adsorbents, which spread out the electrostatic patches, enlarge correlation lengths and reduce voltage dispersions Rossi1992 ().
The pressure due to electrostatic patches between two planes can be computed by solving the Poisson equation Speake2003 (). Its evaluation depends on the spectra describing the correlations of the patch voltages or, equivalently on the associated noise spectra , with a patch wave-vector. In analysis of the patch pressure devoted to Casimir experiments up to recently, the spectrum was assumed to be flat between two sharp cutoffs at a minimum wave-vector and a maximum one . Assuming furthermore that and were given by the grain size distribution measured with an Atomic Force Microscope (AFM), it was concluded that the patch pressure was much smaller than the discrepancy between experiment and theory Decca2005ap (); Decca2007prd (); Decca2007epj ().
A quasi-local model has recently been proposed with the aim of proposing a much better motivated representation of patches Behunin2012pra85 (). The model is based on a tessellation of the sample surface and a random assignment of the voltage on each patch. It produces a smooth spectrum different from the sharp-cutoff model used in previous analysis since there is now contributions to the patch pressure coming from arbitrary low values of , even if the patch size distribution has an upper bound. When the patch effect is estimated with the parameters deduced from the grain size distribution as in Decca2005ap (); Decca2007prd (), a much larger contribution of patches is obtained. In fact, the calculated patch pressure is now larger than the residuals between experimental data and theoretical predictions, which means that patches could be a crucial systematic effect for Casimir force measurements Behunin2012pra85 ().
As the computed patch pressure is model dependent, it seems natural to try to find a model between the two cases presented above which would reproduce at least qualitatively the residuals. By varying the parameters of the quasi-local model, it was found in Behunin2012pra85 () that the output of the model depended mainly on two parameters, the size of largest patches and the rms voltage dispersion and that a best-fit on these two parameters produced a qualitative agreement between the residuals and the patch pressure. The best-fit values for the parameters and are quite different from those obtained by identifying patch and grain sizes. With larger than the maximum grain size and smaller than the rms voltage which would be associated with random orientations of clean micro-crystallites, these values are however compatible with a contamination of metallic surfaces, which had to be expected anyway.
It follows that the difference between IUPUI experimental data and theoretical predictions can be fitted at least qualitatively by a simple model for electrostatic patches. This conclusion is however only the result of a fit, with the parameters of the patch model not measured independently. In order to reach a firm conclusion, the patch spectrum has to be measured independently, by using the dedicated technique of Kelvin probe force microscopy (KPFM) which is able to achieve the necessary size and voltage resolutions Liscio2008jpc (); Liscio2011acr (). When these characteristics are available, the contribution of the patches to the Casimir measurements can be evaluated and unambiguously subtracted when comparing theory and experiments.
Ii A simple derivation of the Casimir effect in one dimension
The present section II contains a derivation of the Casimir effect in a model of scalar fields propagating along the two directions on a 1-dimensional line. Within this simple model which lays the basis for more complicated calculations to appear in the next section, we introduce the Quantum Optics approach to the Casimir effect. The approach is based on the scattering of vacuum field fluctuations obtained in the ground state of the associated Quantum Field Theory. Each mirror is described by a scattering operator Lippmann1950 (); GellMann1953 () which is reduced here to a matrix containing reflection and transmission amplitudes. Two mirrors form a Fabry-Perot cavity with all field transformations deduced from the two elementary scattering matrices. The Casimir force then results from the difference of radiation pressures exerted onto the inner and outer sides of the mirrors by the vacuum field fluctuations. Equivalently, the Casimir free energy can be written as the shift of field energy due to the presence of the Fabry-Perot cavity Schwinger1975 (); Plunien1986 (). The formula obtained in this manner is valid and regular at thermal equilibrium at any temperature and for any optical model of mirrors obeying causality and high frequency transparency properties.
The radiation pressure interpretation of the Casimir force was presented for perfect mirrors in Milonni1988 () and extended to the case of real mirrors Jaekel1991 (). The calculations were then systematically expanded in particular for applications to the problem of the Dynamical Casimir effect Jaekel1992qo (); Jaekel1992jp (); Jaekel1992pla (); Jaekel1993jp1 (); Jaekel1993jp2 (); Jaekel1993pla (); Lambrecht1996prl (); Lambrecht1998epjd (). It has also served as a basis for the Scattering Formalism for the static Casimir effect Genet2003 (); Lambrecht2006 () of which we will give a pedagogical presentation in the following.
ii.1 Quantum field theory on the one-dimensional line
We consider here quantum field theory on the one-dimensional line, that is also quantum field theory in two-dimensional space-time (one time coordinate , one space coordinate ). The field propagation is thus described by the d’Alembert’s wave equation, originally written for the propagation of transverse vibrations of a string, and which also describes many wave phenomena such as electrical propagation in a transmission line, acoustic wave and so on.
Propagation equation on the one-dimensional line
We write it here for a single vibration described by the scalar potential
The general solution to this equation is given by the d’Alembert’s formula, that is the superposition of rightward () and leftward () traveling waves propagating at the velocity in opposite directions along the -axis
where are called today the light cone variables
In this simplest version of field theory, there is one normal mode for each frequency and each propagation direction . The standard methods of quantum field theory Itzykson1985 (); Cohen-Tannoudji1992 () then allow one to write the rightward () and leftward () traveling waves as Fourier decompositions over canonical mode operators
Annihilation and creation operators and correspond respectively to positive and negative frequencies in the decompositions (20). Note that the fields in space-time are real-valued, so that annihilation and creation operators are hermitian conjugate of each other. They obey the following canonical commutation relations
The Hamiltonian for the d’Alembert’s wave equation (17) is the integral over space of the energy density
Using the d’Alembert’s formula (18) for fields, one derives another d’Alembert’s formula describing the general energy density as the superposition of rightward and leftward traveling flows
Vacuum and thermal fluctuations
Vacuum is the fundamental state of quantum field with an infinite number of modes each containing no photons. This means that all annihilation operators vanish in this state while creation operators have their action determined by the commutation relations (II.1).
In the following, we use the correlation functions of vacuum fields which are deduced from these elementary properties
where . Higher-order correlation functions are deduced from the fact that vacuum fields may be dealt with as Gaussian random variables.
In a state at thermal equilibrium at temperature , the first and last lines in (24) are unchanged whereas the second and third lines are changed to
where while is the mean photon number in Planck’s law (2). Note that (25) is reduced to (24) when . Note also that all field commutators are unchanged, which is consistent with the fact that they are directly connected to the propagators.
Using the Fourier decompositions (20) of the fields and the correlation functions (25), we deduce that the mean values of energies densities (23) have the following spectral decompositions in the general case ()
This is the modern expression of the second Planck’s law Planck1911 () for the energy per mode given as the sum of vacuum and thermal contributions (compare with (2)). The limit of zero temperature corresponds to in this expression.
ii.2 One mirror on a 1d line
We now consider the situation where one mirror is placed into vacuum at a position (see Fig.5).
Scattering by one mirror on a 1d line
The general solution is in this case a generalized d’Alembert’s formula with different expressions for fields on the lefthand () and righthand () sides of the mirror
The symbol and represent as previously rightward and leftward propagation while the symbols in and out correspond to incoming and outgoing traveling waves. These waves are coupled by scattering on the mirror.
For the limiting case of perfectly reflecting mirrors, the field vanishes at the left and right sides of the mirrors. Output fields are then easily deduced from input ones as . In Fourier space, this is written as a scattering process with reflection amplitudes having a unit modulus and a phase determined by the position of the mirror, with . As the scattering process preserves the frequency for a motionless mirror, these equations could as well have been written for annihilation () and creation () operators.
Real mirrors on a 1d line
Real mirrors cannot be perfectly reflecting at all frequencies. They are described by a more general scattering matrix containing transmission as well as reflection amplitudes (with )
The scattering amplitudes and have been defined for a mirror located at and the general case then obtained by introducing the phases determined by the position of the mirror. We have considered the particular case of a symmetrical scattering matrix for a mirror located at . A more general treatment would not change any important result in the following.
The scattering matrix preserves frequency since energy is conserved for a stationary scattering but it depends on frequency as a consequence of fundamental physical properties Jaekel1991 (). The scattering process considered here obeys the following properties (the unitarity assumption, valid only for lossless mirrors, will be released later on)
Fields are real in the time domain, so that , that is also and ;
The scattering process obeys causality, so that the amplitudes can be prolongated as analytical functions in the upper half of the complex plane (more details below);
The scattering process obeys unitarity, so that , that is also and ;
Reflection tends to vanish at the high-frequency limit , so that and .
These general properties may be illustrated with an example, which corresponds in particular to a transmission line with a localized impedance mismatch Jaekel1993pla (). For this simple example, the d’Alembert’s wave equation (17) is changed to
and the solution obtained as (28) with
The general properties (1-4) enumerated in the preceding paragraph can easily be checked out. The parameter appears as the physical cutoff above which reflection tends to vanish. The limit of perfect reflection can be defined by the case of large values of . It has been shown that this definition allows one to escape the difficulties encountered when studying perfectly reflecting mirrors Jaekel1992pla (); Jaekel1993jp1 ().
Force on one mirror on the 1d line
The general solution is now a d’Alembert’s formula (23) for energy densities with different expressions on the lefthand () and righthand () sides of the mirror
Similar expressions can be written for the momentum densities by just putting a sign in front of energy densities . The force on the mirror is then deduced from a momentum balance upon scattering and it is found to be proportional to the difference of the energy densities and on the lefthand and righthand sides of the mirror Jaekel1992qo ().
In the limiting case of perfectly reflecting mirrors, the field vanishes at the left and right sides of the mirrors. Output fields are then easily deduced from input ones as , and the force thus obtained as . The mean values of leftward and rightward energy densities are infinite (see (26)). But these mean values are also equal for leftward and rightward densities, so that the force on the mirror is zero .
This result crucially depends on the fact that we have evaluated the mean force on a mirror at rest. There exist non vanishing fluctuations of force on the mirror, as well as a non null mean force for a moving mirror Jaekel1992qo ().
Force on one real mirror
In the general case of a non perfect mirror, the same result is proven by the following reasoning. First, the force is deduced from the momentum balance upon scattering and the expressions (31) of energy densities
Then, the mean energy densities are calculated by using the expressions (23) of energy densities and the description (28) of the scattering process. From the unitarity of the matrix, one deduces that mean radiation pressures are still equal on the two sides (equation written at thermal equilibrium at temperature )
so that the mean value of the force still vanishes
ii.3 Two mirrors on the 1d line
The situation changes fundamentally as soon as we consider that there are two mirrors present on the 1d line, which form a Fabry-Perot cavity. There is now a difference between the inner and outer sides of the mirrors, as sketched on Figure 7.
Scattering by a cavity on a 1d line
The spatial positions of the two mirrors are denoted and with the length of the cavity . Each mirror couples rightward and leftward traveling waves. In contrast with the case of one mirror, the fields undergo multiple scattering and there appears an intra-cavity region. The general solution for the fields is now written with different expressions for fields on the lefthand () and righthand () sides of the cavity and those () within the cavity
The scattering effect of the cavity has now to be described by two matrices, instead of one in the one mirror case, a global scattering matrix which gives the output fields in terms of the input ones, and a resonance matrix which gives the intra-cavity fields
Scattering and resonance matrices
The global matrix can be evaluated from the elementary matrices and associated with the two mirrors and the free propagation phase-shifts. The results can be written as follows (all amplitudes depend on )
The denominator is an important function, with its zeros corresponding to the resonances of the cavity,
For the problem under consideration, the global matrix obeys the same properties as the elementary matrices and . In particular, it is unitary
The resonance matrix can be deduced from the elementary matrices and associated with the two mirrors and the free propagation phase-shifts. The results are obtained as (amplitudes depend on and expressions are simplified by assuming )
The resonance matrix shares some of the properties listed above with the matrix but it is not unitary. It turns out that can be written Jaekel1992jp ()
ii.4 Casimir force on the 1d line
We now evaluate the energy densities and the forces in the case of two mirrors on the 1d line.
We deduce expressions for the forces acting on each mirror 1 or 2, obtained as differences of the radiation pressures on the lefthand and righthand sides of the mirror (compare with (II.2))
The energy densities on the outer sides of the cavity have the same expressions as in the one-mirror case, provided that the scattering matrix (LABEL:globalSmatrix) of the cavity is used. As this global matrix is unitary (see (39)), one deduces that the mean energy densities on the outer sides are still given by equations (33). Hence, they are infinite but equal on the two sides of the cavity so that the mean value of the global force on the cavity vanishes
Intra-cavity energy densities
The situation is different for the energy densities on the intra-cavity sides of the mirrors, since their calculation is now determined by the properties of the resonance matrix. Using the property (41), one deduces that the mean values of these energy densities is given by Jaekel1992jp ()
where is the common value of the diagonal elements in the matrix
The real function represents the modification of the energy density inside the cavity with respect to that outside the cavity, which is in fact identical for input and output fields. The same result would have been obtained for a classical calculation with an input field at frequency . Here this function describes the change of energy densities for vacuum as well as thermal fluctuations. Its relation (46) to the closed loop function will be used in the following to transform the expression of the Casimir force.
Casimir force as a result of radiation pressures
Collecting these results, one deduces the following expression of the Casimir force, defined as the mean force on the righthand mirror or the opposite of that on the lefthand mirror (see (44)),
Resonant frequencies correspond to an increase of energy in the cavity () and they produce repulsive contributions to the Casimir force. In contrast, frequencies out of resonance correspond to a decrease of energy in the cavity () and they produce attractive contributions to the Casimir force. The net force is the integral of these contributions over all modes. This interpretation of the Casimir force as a result of radiation pressures of vacuum and thermal fluctuations produces a final expression which is finite for any properly defined model of mirrors Jaekel1991 (). This is seen more easily by using causality properties to rewrite (47) as an integral over imaginary frequencies. Let us stress at this point that this rewriting is just a mathematical transformation which does not affect the physical content of (47). The rewriting will however spoil its direct intelligibility as imaginary frequencies do not correspond to physical modes.
Casimir force as an integral over imaginary frequencies
One now rewrites the Casimir force as an integral over imaginary frequencies by using the causality properties of the scattering amplitudes. We give here a simplified description (a more general derivation can be found in Guerout2014 ()).
We first write the Casimir force (47) as the real part of a complex integral defined over the positive part of the real axis
We have used the relation (46) between and , substituted the frequency by a complex variable running over . We have also replaced by its explicit form (4). Now the closed loop function is defined from causal reflection amplitudes and propagation phases. Considered as a function of , it has poles in the lower half part of the complex plane which correspond to resonances of the Fabry-Perot cavity, while it is analytical in the upper half part of the complex plane . Meanwhile the function is analytical in the right half part of the complex plane but has poles at the Matsubara frequencies which are regularly spaced on the imaginary axis Matsubara1955 ()
It follows that the integral can be transformed by using Cauchy’s theorem. Precisely, we apply the Cauchy’s theorem to the integral of the integrand appearing in (48) over a closed contour consisting of , the positive part of the imaginary axis shifted by a small positive real number and a quarter of a circle with a very large radius. This last part vanishes as a consequence of the high frequency transparency of the mirrors. The integral over the whole contour also vanishes since the integrand is an analytical function in the domain enclosed by the contour. As a consequence, the integral may be written under the equivalent form ()
The same transformation is then performed for which is the integral of the same function over the negative part of the real axis, run from to 0. is equal to , the integral of the same integrand over the positive part of the imaginary axis shifted by a small negative real number and run from to .
In the end, the Casimir force (48) is the integral over a contour which encircles the imaginary axis, and it is thus found to be a discrete sum of the values of the function at the Matsubara poles
The primed sum symbol implies that the contribution of the zeroth Matsubara pole at is counted for only one half (symbol written here for a function )
This final expression is always finite for any properly defined model of mirrors.
In the limit , the ensemble of Matsubara poles becomes a cut along the imaginary axis (the function thus goes to +1 for and to -1 for ). The discrete sum (51) is then written as an integral over the positive part of the imaginary axis
For perfect mirrors, i.e. when may be taken as unit value at all frequencies contributing to the integral (53), a universal result is obtained, which no longer depends on the specific properties of the mirrors
We have used the fact that the Riemann zeta function
takes the value for . Other values of the same function will appear in various places in Casimir force calculations with perfectly reflecting mirrors.
For real mirrors the product of reflection amplitudes always has a modulus smaller than unity. It follows that the integral is always regular with a smaller modulus than for perfect mirrors. If the two mirrors are identical (), then their product is positive and the force is attractive as in the case of perfect mirrors. The expression of the Casimir force as the integral (54) over imaginary frequencies is convenient to discuss the meaning of the limit of perfect mirrors. Taking as an example the model (30), we indeed see that (53) tends to (54) as soon as the characteristic frequency at which the reflection falls down is larger than the typical frequencies contributing to the integral.
ii.5 The Casimir free energy and phase-shift interpretation
We now write an expression for the Casimir free energy and show that it can be given an interpretation in terms of scattering phase-shifts. We also obtain expressions for the Casimir entropy and Casimir internal energy.
The Casimir free energy
We come back to the expression (48) of the force as an integral over real frequencies and write it as the differential of a free energy with respect to
We show in the following that this formula can be given a nice interpretation in terms of phase-shifts Jaekel1991 (). Note that, though could be changed by a independent contribution without changing the result for , the form of given in (II.5) is fixed by the phase-shift interpretation discussed below.
The phase-shift interpretation
In the absence of the scatterer, the modes in the large box would have their wavelengths determined by the box size through , where labels the rightward and leftward propagation directions which correspond to degenerate solutions. The eigen-modes in the presence of the scatterer are thus determined by the eigenvalues of the unitary matrix through . The change of global energy of the modes at a given frequency is then determined by the quantity proportional to the sum of the two phase-shifts at this frequency. As a consequence, we do not need to solve the full eigenvalue problem for the matrix, but we need only to calculate the logarithm of its determinant .
The relation between the expression (II.5) of the free energy and the phase-shift interpretation is then fixed by noting that the expression (LABEL:globalSmatrix) written above for the matrix associated to the cavity leads to the following relation between the determinants
where we have denoted the scattering matrix for the compound system consisting of the two mirrors 1 and 2 (it was simply denoted in (LABEL:globalSmatrix)). One thus deduces
It is now clear that the Casimir free energy (II.5) is given by a difference between changes of free energies calculated in three different scattering configurations
with each of these quantities determined by the phase-shifts for the associated matrix
Similar expressions hold for the change of free energies and associated respectively with mirrors 1 and 2 placed in the large box on Figure 8.
In fact, each of the changes is itself a difference of free energies calculated in the presence and in the absence of the scatterer
with the subscript 0 labeling the configuration with no scatterer. In the end, the Casimir free energy is a difference involving four different configurations