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Casimir forces

Casimir effect and vacuum fluctuations

In its modern quantum conception, space is filled by vacuum field fluctuations which have physical consequences such as radiative corrections or spontaneous emission. They have also mechanical effects in macro- and micro-physics, called the Casimir and Casimir-Polder forces respectively.

Measurements of Casimir force reach a good experimental precision, and the comparison with theoretical predictions has interesting connections with open questions in fundamental physics. Casimir physics is related to the still unsolved problem of vacuum energy and it plays an important role in the tests of gravity at short ranges.

The notes of the lectures at the 101th Les Houches Summer School “Quantum Optics and Nanophotonics” present the main features of the theory of Casimir forces and its comparison to experiments. They contain references on many topics of interest for this domain.

  • Casimir forces, S. Reynaud, A. Lambrecht, in Quantum Optics and Nanophotonics, Fabre C., Sandoghdar V., Treps N. and Cugliandolo L. eds (Oxford University Press, 2017) pp. 407-455 [arXiv version]
  • Questions can be sent to Serge Reynaud

In the so-called scattering approach developed by our team, the Casimir energy is the change of vacuum energy induced by the coupling of arbitrary objects with electromagnetic fluctuations. This coupling is described by scattering matrices containing reflection and transmission amplitudes depending on field modes.

This scattering method can be applied to the calculation of Casimir forces between macroscopic mirrors with various geometries (plane, spherical, cylindrical, nanostructured…) as well as Casimir-Polder forces for atoms/molecules near a surface or other atoms/molecules.

Mirrors used in precise Casimir experiments are metallic and the Casimir force is affected by their finite conductivity. The scattering approach allows in principle to describe these effects but important questions remain open for the treatment of dissipative media coupled to electromagnetic fluctuations at non zero temperature.

Most accurate experiments measure the force between a plane and a sphere in electromagnetic fields. With the scattering formula developed on adapted wave basis, expressions are obtained for arbitrary values of the sphere radius, inter-plate distance and material properties. These calculations go beyond the proximity force approximation, which has been used for analyzing most experiments.

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

Casimir-Polder forces and quantum reflection

Quantum reflection (QR) is a reflection for matter waves in a rapidly varying attractive potential. It is a counter-intuitive quantum effect observed for atoms feeling the attractive Casimir-Polder (CP) potential in the vicinity of a surface.

Our team has studied this effect for antihydrogen atoms in the context of the GBAR project which aims at measuring their free fall in the Earth gravity field (described on another page on this site). As quantum reflection may prevent annihilation of antihydrogen atoms on surfaces on which they should be detetcted, taking the effect into account is absolutely necessary in the data analysis.

  • Quantum reflection of antihydrogen from the Casimir potential above matter slabs, G. Dufour, A. Gérardin, R. Guérout, A. Lambrecht, V.V. Nesvizhevsky, S. Reynaud, A.Yu. Voronin Phys. Rev. A 87 012901 (2013)

Quantum reflection depends on the magnitude of the potential and on the steepness describing its rate of variation, which makes the dependence on parameters far from intuitive. QR probability increases when the energy of the incident atom is decreased, or when the absolute magnitude of the CP potential is decreased. Our team has developed a method based on Liouville transformations to obtain a better understanding of these paradoxical behaviors.

Liouville transformations map in a rigorous manner one Schrödinger solution into another with a modified potential. Scattering properties are invariant under the transformation although the semiclassical descriptions are completely different. For example quantum reflection of an atom on an attractive CP well can be changed into reflection of the atom on a repulsive wall.

The paradoxes in the initial problem become intuitive predictions in the transformed problem. Furthermore, a qualitative understanding of the QR probability can be deduced from the exact solution known ffor the far-end Casimir–Polder potential (behaving as the inverse fourth power of distance).

  • Quantum reflection and Liouville transformations from wells to walls, G. Dufour, R. Guérout, A. Lambrecht, S Reynaud, EPL 110 30007 (2015)
  • Liouville transformations and quantum reflection, G. Dufour, R. Guérout, A. Lambrecht, S. Reynaud, J. Phys. B 48 155002 (2015)

Quantum reflection from the attractive Casimir-Polder interaction holds atoms against gravity and traps them in quantum levitation states. The latter can be viewed as trapped states for ultracold atoms in a cavity with gravity and Casimir-Polder potentials acting, respectively, as top and bottom mirrors.

Casimir-Polder shifts of the cavity resonances and associated lifetimes can be calculated numerically or represented by analytical expressions (improved effective range expansion) which are accurate enough for quantum techniques aimed at tests of the weak equivalence principle on antihydrogen.

Casimir attraction at the cell scale

In collaborations with colleagues, our team has shown that long-range interaction between objects in an ionic fluid, via electromagnetic field fluctuations, may be important for understanding the self-organization at the cell scale.

Biological fluids are media where charges abound, and it is commonly accepted that electromagnetic interactions have a short effective range, due to the screening at long ranges: thanks to a spatial organization of charges of opposite signs, the medium appears at mesoscopic scales as essentially neutral, and forces of electrostatic origin are weak.

Using numerical simulations, our work has shown that some electromagnetic modes (the transverse field modes) propagate in the ionic liquid as they are not affected by screening. Fluctuations in these modes are responsible for a long-range attractive interaction, a Casimir force .

The Casimir force exists between two dielectric spheres in salt water but, in this geometry, the interaction can only exceed the energy of thermal fluctuations present in the liquid when the spheres are very close to each other. The force has been measured in an experiment in the optical tweezers laboratory at the University of Rio de Janeiro (Brazil)

  • Probing the screening of the Casimir interaction with optical tweezers, L. B. Pires, D. S. Ether, B. Spreng, G. R. S. Araújo, R. S. Decca, R. S. Dutra, M. Borges, F. S. S. Rosa, G.-L. Ingold, M. J. B. Moura, S. Frases, B. Pontes, H. M. Nussenzveig, S. Reynaud, N. B. Viana, and P. A. Maia Neto; Phys. Rev. Research 3, 033037 (2021)

The force has been shown theoretically to depend on geometrical parameters (radius of each sphere and separation between them) through universal laws

  • Universal Casimir Interaction between Two Dielectric Spheres in Salted Water, T. Schoger, B. Spreng, G.-L. Ingold, P. A. Maia Neto, and S. Reynaud; Phys. Rev. Lett. 128, 230602 (2022)

In a collaboration with colleagues at Gulliver laboratory in ESPCI, UC Davis (USA), Jülich (Germany), EPFL Lausanne (Switzerland) and the University of Rio de Janeiro (Brazil), we have studied the geometry of dielectric cylinders in salt water, and shown that the Casimir interaction has a much larger magnitude in this case, as it is proportional to the length of the cylinders. The binding energy between actin filaments grouped together in bundles within the cell thus exceeds the energy of thermal fluctuations in this biologically relevant situation.

This interaction exhibits universality properties as it does not depend on the detailed dielectric properties of the filaments and the solvent, making the results of this work applicable to multiple configurations of biological or physico-chemical interest involving cylindrical structures. The long range of Casimir interactions is in particular expected to have important effects on the cohesion and self-assembly of filamentary structures of biological media at cell scale.

  • “Universal Casimir attraction between filaments at the cell scale”, B. Spreng, H. Berthoumieux, A. Lambrecht, A-F. Bitbol, P. Maia Neto, et S. Reynaud, New J. Phys. 26, 013009 (2024)
  • Questions can be sent to Serge Reynaud
Casimir binding energy is shown as a function of separation d (nm) between two actin filamentseach having a radius of 3 nm and length of 15 μm. Binding energy |F| is measured in units of thermal energy kBT, and the shaded zone corresponds to |F| < kBT. Energies above the shaded zone should play important roles, and this is the case for the numbers for actin bundles in cells, with a separation of 6 nm (red disk on figure). The inset shows two filaments in an actin bundle (blue) and cross-linkers between them (red) © B. Spreng et. al. et The New Journal of Physics (licence CC-BY 3.0).

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