LKB - Quantum fluctuation and relativity


Jean-Michel Courty
Marie-Pascale Gorza
Romain Guérout
Astrid Lambrecht
Serge Reynaud

In its modern quantum conception, space is filled by vacuum field fluctuations which have observable consequences in microscopic physics, for example the radiative corrections or the spontaneous emission processes. These vacuum fluctuations also have observable mechanical effects in nano- and mesoscopic physics, in particular the Casimir force between two mirrors. Measurements of this force now reach a good experimental precision, but the comparison with theoretical predictions raises persisting difficulties.

This comparison is interesting because of its interfaces with open questions in fundamental physics. The static Casimir effect is connected with the puzzles of gravitational physics through the problem of vacuum energy while the dynamic Casimir effect shows the limit of the principle of relativity of motion associated with the classical theory of relativity. Casimir physics also plays an important role in the tests of gravity at short ranges.

The notes of the lectures given at the Les Houches Summer School “Quantum Optics and Nanophotonics” in August 2013 contain a description of the current state-of-the-art for Casimir force measurements as well as 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.

Casimir effect in the scattering formalism

Schematic representation of the Casimir force as the result of vacuum radiation pressure summed over all field modes.

We have developed the scattering method which allows one to describe the dispersive interaction between arbitrary objects. The coupling of these objects with the electromagnetic field is described by scattering matrices containing reflection and transmission amplitudes which depend on frequency, wave-vector and polarization.

The Casimir energy is the change of vacuum energy while the Casimir force can be obtained as a sum over all field modes of the radiation pressure of vacuum fluctuations.

The scattering method can be applied to the calculation of the Casimir force between two mirrors (plane, spherical, nanostructured or rough) as well as that of the Casimir-Polder interaction for an atom or a molecule near a surface in the evaluation of van der Waals forces between atoms, molecules or clusters.

The scattering approach was applied to the static Casimir effect with an increasing degree of generality in the following papers:

Material properties and Casimir effect

Common configuration for the calculation of Casimir force : plane-plane geometry between two gold-covered mirrors.

Mirrors used in most experiments are metallic and the Casimir force is affected significantly by their finite conductivity, in a way that depends on the distance between the mirrors. Large distances correspond mainly to low frequencies for which the metallic mirrors are almost perfectly reflecting, so that the Casimir force should approach the force calculated for ideal mirrors. This approach however shows subtle properties for dissipative media at non zero temperature.

  • Quantum dissipative Brownian motion and the Casimir effect, G-L. Ingold, A. Lambrecht, S. Reynaud, Phys. Rev. E. 80, 041113 (2009)
  • Derivation of the Lifshitz-Matsubara sum formula for the Casimir pressure between metallic plane mirrors, R. Guérout, A. Lambrecht, K.A. Milton, S. Reynaud, Phys. Rev. E. 90, 042125 (2014)

Recent experiments have been performed between magnetic metallic plane mirrors. As in the case of nonmagnetic materials, the commonly used expression is valid for the lossy model of optical response, but not for the lossless plasma model. In fact, the modes associated with the Foucault currents play a crucial role in the limit of vanishing losses, in contrast to expectations.

  • Lifshitz-Matsubara sum formula for the Casimir pressure between magnetic metallic mirrors, R. Guérout, A. Lambrecht, K.A. Milton, S. Reynaud, Phys. Rev. E. 93, 022108 (2016)

For (non-magnetic) metallic mirrors at the short distances (typically below one micron) associated with high-frequency modes, the metallic mirrors behave as poor reflectors and the effect of finite conductivity reduces the Casimir force between the mirrors.

These theoretical results show the importance of a full comprehension of the influence of the material properties of the mirrors on the Casimir forces. In collaboration with experimental groups, we investigated these effects under different conditions.

  • Quantitative non contact dynamic Casimir force measurements, G. Jourdan, A. Lambrecht, F. Comin, J. Chevrier, EPL 85, 31001 (2009)
  • Influence of slab thickness on the Casimir force, I. Pirozhenko and A. Lambrecht, Phys. Rev.A 77, 013811 (2008)
  • Casimir force between a metal and a semimetal, G. Torricelli, I. Pirozhenko, S. Thornton, A. Lambrecht, C. Binns, EPL 93 (2011) 51001
  • Measurement of the Casimir effect under ultrahigh vacuum: Calibration method, G. Torricelli, S. Thornton, C. Binns, I. Pirozhenko, and A. Lambrecht J. Vac. Sci. Technol. B 28, C4A30 (2010)

At very short distances, the Casimir force corresponds directly to the instantaneous interaction between the surface plasmons living on the two metallic surfaces. Surface plasmons produce a repulsive contribution to the force for median and large distances, while the global force remains attractive. It can thus be expected that the Casimir forces may be affected by changing the coupling between plasmons and photons through nanostructured surfaces.

- Surface plasmon modes and the Casimir energy, F. Intravaia, A. Lambrecht, Phys. Rev. Lett. 94 (2005) 110404
- The Role of Surface Plasmons in The Casimir Effect, F. Intravaia, C. Henkel, A. Lambrecht, Phys. Rev. A 76 (2007) 033820
- Repulsive Casimir forces and the role of surface modes, I. Pirozhenko, A. Lambrecht, Phys. Rev; A 80 (2009) 042510

Casimir forces in the plane-sphere geometry

Common configuration for measurements of Casimir force : plane-sphere geometry between two gold-covered mirrors.

The Casimir energy has to be computed in the geometry of interest for the most precise experiments, a plane and a sphere in electromagnetic vacuum or in a thermal field. To this aim, the scattering formula can be developed on adapted plane-wave and spherical-wave basis, leading to an expression valid for arbitrary values of the sphere radius, inter-plate distance and plasma wavelength. These calculations go beyond the proximity force approximation, where contributions to the Casimir force at different distances are summed up as if they were independent.

With this method, we were able to produce calculations of the Casimir force, taking into account for the first time geometry effects as well as material properties of the plates (Gold in most experiments).

  • Casimir Interaction between Plane and Spherical Metallic Surfaces, A. Canaguier-Durand, P.A. Maia Neto, I. Cavero-Pelaez, A. Lambrecht, S. Reynaud, Phys. Rev. Lett. 102, 230404 (2009)
  • Casimir energy between a plane and a sphere in electromagnetic vacuum, P.A. Maia Neto, A. Lambrecht, S. Reynaud, Phys. Rev. A 78 012115 (2008)

In the case of a non zero temperature, we did calculate a negative interaction entropy and show that the contribution of thermal photons to the Casimir force could be repulsive. The results of the calculation also allowed us to estimate precisely the deviation from previous evaluations made within the proximity approximation. The electromagnetic result was found to depart from this commonly used approximation significantly more rapidly than expected from existing scalar computations.

In the classical limit of high temperatures we found that the Casimir interaction in the plane-sphere geometry between two Drude metals becomes independent of the metal’s conductivity, which is not the case if the material properties are modeled by a plasma model. The Drude model has thus a more universal character than the plasma model. Another very interesting result of this study is that development in the vicinity of the local approximation can not be represented by an analytical series in the aspect ratio that characterizes the geometry.

  • Classical Casimir interaction in the plane-sphere geometry, A. Canaguier-Durand, G.-L. Ingold, M.-T. Jaekel, A. Lambrecht, P.A. Maia Neto, S. Reynaud, Phys. Rev. A 85, 052501 (2012)


Theory-experiment comparison

The Casimir effect is an important prediction of Quantum Field Theory, which deserves a careful experimental verification. The comparison between experimental measurements and theory is also important as a tool to derive constraints on hypothetical new short-range interactions appearing in addition to the gravity force in unification models. A review bearing on gravity tests at sub-mm scales can be found in

  • Short-range fundamental forces, I. Antoniadis, S. Baessler, M. Buechner, V.V. Fedorov, S. Hoedl, A. Lambrecht, V.V. Nesvizhevsky, G. Pignol, K.V. Protasov, S. Reynaud, Y. Sobolev, C. R. Phys. 12 755-778

The comparison between experiments and theory has to account for the many differences between the ideal case initially discussed by Casimir and the accurate experiments performed in the last years. These experiments are performed with imperfectly reflecting mirrors (not perfect ones), in the plane-sphere geometry (not in the more easily calculated plane-plane geometry), and at room temperature (the effect of thermal fluctuations has to be taken into account besides that of vacuum fluctuations). Meanwhile, the surfaces of the mirrors are not perfect as they show roughness (topographic irregularities) and patches (voltage irregularities). We have worked on many of these effects in an effort to assess the significance of the unexpected results obtained from this theory-experiment comparison (some of them are discussed in the neighboring pages on this site).

Casimir force measurements performed at distances smaller than 750 nm at IUPUI and UCR were interpreted by the authors as excluding the dissipative Drude model and agreeing with the lossless plasma model. In distinction, a recent experiment at Yale was able to measure Casimir forces at distances up to 7 μm and was interpreted by the authors as being in agreement with the Drude prediction, once an electrostatic patch contribution was taken into account.


Distribution of the irregular patch potentials measured by KPFM on gold samples similar to those used in Casimir experiments.

It is known that patch effects are a source of concern for Casimir experiments as well as for other precision measurements. Unfortunately, it was not possible in any of these experiments to measure the patches independently. It follows that the conclusions of the theory-experiment comparisons heavily rely on the patch models used in the data analysis. In collaboration with colleagues at Los Alamos National Laboratory and UF Rio de Janeiro, we have analyzed the patch contribution to Casimir experiments with a model of quasi-local voltage correlations. This model, which is derived from well-motivated physical principles, shares key features with models used to describe heating in ion traps. We have shown that this model could be used to fit the difference between theory and the experimental data of the IUPUI experiment. At this stage however, it was only a fit, not an explanation of the discrepancy.

  • Modeling electrostatic patch effects in Casimir force measurements, R. O. Behunin, F. Intravaia, D. A. R. Dalvit, P. A. Maia Neto, S. Reynaud Phys. Rev. A 85, 012504 (2012)

In order to solve this issue, Kelvin probe force microscopy was performed by two different groups on the Au-coated planar samples which were used to measure the Casimir interaction at IUPUI. The obtained voltage distribution was used to calculate the separation dependence of the patch pressure in the plane-sphere geometry of these Casimir measurements. The result of this calculation, using the currently available knowledge, is that the patch pressure does not explain the magnitude or the separation dependence of the difference between the measured Casimir pressure and the one calculated using a Drude model.

  • Kelvin probe force microscopy of metallic surfaces used in Casimir force measurements, RO Behunin, DAR Dalvit, RS Decca, C Genet, IW Jung, A Lambrecht, A Liscio, D Lopez, S Reynaud, G Schnoering, G Voisin and Y Zeng Phys. Rev. A 90, 062115 (2014)
  • Electrostatic patch effects in Casimir-force experiments performed in the sphere-plane geometry, R.O. Behunin, Y. Zeng, D. A. R. Dalvit, S. Reynaud Phys. Rev. A 86, 052509 (2012)
Casimir forces and nanophysics

Schematic representation of a configuration with a nano-corrugated plate facing a plane one.

Nano-structured surfaces have gained considerable interest in nanophysics. The effects of thermal and quantum fluctuations, in particular through Casimir-like forces, play an important role at the scale of these systems.

The scattering approach allows one to calculate the Casimir interaction energy between arbitrary objects. This method was first applied to evaluate the force between periodic dielectric gratings within the project ANR Pnano MONACO with the LETI/CEA.

In contrast to the case of specular reflection on a plane, scattering on a grating changes the wave-vector and polarization of the field. As the scattering couple different diffraction orders, the reflection and transmission matrices are obtained by solving a system of coupled differential equations the number of which is proportional to the number of diffraction orders that need to be considered.

This method has allowed us to compare with theory the Casimir force measurements performed between a Silicon grating and a Gold sphere in the group of H.B. Chan (Phys. Rev. Lett. 101, 030401, 2008 & Phys. Rev. Lett. 105, 250402, 2010).

  • Casimir interaction between dielectric gratings, A. Lambrecht, V.N. Marachevsky, Phys. Rev. Lett. 101, 160403 (2008)
  • Casimir Force on a Surface with Shallow Nanoscale Corrugations : Geometry and Finite Conductivity Effects, Y. Bao, R. Guérout, J. Lussange, A. Lambrecht, R. A. Cirelli, F. Klemens, W. M. Mansfield, C. S. Pai, H. B. Chan, Phys. Rev. Lett. 105, 250402 (2010)
  • Thermal Casimir force between nanostructured surfaces, R. Guérout, J. Lussange, H. B. Chan, A. Lambrecht, S. Reynaud, Phys. Rev. A 87, 052514 (2013)

Above nano-structured surfaces, the translational invariance of vacuum in transverse direction is broken and a lateral Casimir force arises. It is often calculated through a simple approximation (Proximity Force Approximation or PFA) where force contributions at different distances are added as if they were independent from each other, though this approximation is not valid in general. In particular it leads to wrong results for corrugated surfaces with corrugation wavelength smaller than the other relevant length scales.

  • Lateral Casimir force beyond the proximity force approximation: A nontrivial interplay between geometry and quantum vacuum, R. B. Rodrigues, P. A. Maia Neto, A. Lambrecht, S. Reynaud, Phys. Rev. A 75, 062108 (2007)

Schematic representation of a configuration with two nano-corrugated plates facing each other. When the corrugations are not aligned, a Casimir torque arises.

When two corrugated metallic plates are placed in vacuum, their interaction with electromagnetic vacuum fields produce a torque which tends to align the corrugations. This Casimir torque could be measured with torsion pendulum techniques for separation distances as large as 1$\mu$m. It would thus allow one to probe the nontrivial geometry dependence of the Casimir energy in a configuration which can be evaluated theoretically with accuracy. In the optimal experimental configuration, the commonly used proximity force approximation turns out to overestimate the torque by a factor 2 or larger.

  • Casimir torque between corrugated metallic plates, R. B. Rodrigues, P. A. Maia Neto, A. Lambrecht, S. Reynaud, J. Phys. A 41 164019 (2008)
  • Vacuum-induced torque between corrugated metallic plates, R.B. Rodrigues, P.A. Maia Neto, A. Lambrecht, S. Reynaud, Europhys. Lett. 76, 822 (2006)
  • Casimir torque between nanostructured plates, R. Guérout, C. Genet, A. Lambrecht, S. Reynaud, EPL 111, 44001 (2015)
Quantum thermodynamics and the Casimir effect

The calculation of Casimir effect at non-zero temperature involves the effect of thermal fluctuations (often considered as classical) besides that of quantum vacuum fluctuations. In the plane-plane geometry, dissipative mirrors described by the Drude model give rise to a negative interaction entropy which does not appear with the non-dissipative plasma model.

Ratio of Casimir force at room temperature to the zero temperature force, computed between perfectly reflecting sphere and plane. Solid lines from bottom to top correspond to increasing values of sphere radii. The upper dashed curve represents the PFA expression while the lower dotted curve is the analytical asymptotic expression for L≫R.

In the sphere-plane geometry we have shown that the Casimir interaction may also produce negative entropy even for perfectly reflective mirrors. This is seen on the figure as the fact that the force can be smaller at room temperature than at zero temperature. The proximity force approximation (dashed line on the figure), based on a calculation of the force in the plane-plane geometry, does not match this effect which is intrinsically linked to geometry.

  • Thermal Casimir effect for Drude metals in the plane-sphere geometry A. Canaguier-Durand, P.A. Maia Neto, A. Lambrecht, and S. Reynaud, Phys. Rev. A 82, 012511 (2010)

The latter effect is most pronounced in the dipole approximation, which is reliable when the size of the sphere is small compared to the separation between the sphere and the plate. We have shown that negative entropy can occur in many different situations: between two electrically and magnetically polarizable nano-particles or atoms, which need not be isotropic, and between such a small object and a conducting plate; between two perfectly conducting spheres, between two electrically polarizable nano-particles if there is sufficient anisotropy, between a perfectly conducting sphere and a Drude sphere, and between a sufficiently anisotropic electrically polarizable nano-particle and a transverse magnetic conducting plate.

  • Negative Casimir entropies in nanoparticle interactions, K.A. Milton, R. Guérout, G.-L. Ingold, A. Lambrecht, S. Reynaud, J. Phys. Condensed Mat. 27, 214003 (2015)
  • Geometric origin of negative Casimir entropies: A scattering-channel analysis, G.-L. Ingold, S. Umrath, M. Hartmann, R. Guérout, A. Lambrecht, S. Reynaud, K.A. Milton, Phys. Rev. E 91, 033203 (2015)