We study methods [1,2] for the manipulation of the force of the quantum vacuum known as the Casimir force. It is possible to turn the Casimir force from attraction to repulsion and to use it for levitating mirrors on, literally, nothing. This research may be interesting for applications in nanotechnology, because the Casimir force is the ultimate source of friction for micro- and nano-machines. In the following we explain the science behind Quantum Levitation [1]. See also the article Perfect lens could reverse Casimir force in PhysicsWeb.

A gecko can hang on a glass surface using only one toe. This extraordinary feat of their extraordinary feet is due to the forces between the glass and the gecko's toes, the forces between neutral atoms or molecules known as the van der Waals force [3]. The van der Waals force usually causes things to stick; the force is attractive; and it acts only across short ranges.

  Gecko feet [3].

Field lines of the van der Waals force between two atoms or molecules. From a lecture course at MIT.

What is the van der Waals force? Although a neutral atom or molecule is not electrically charged in total, the charges in the molecule may separate, forming a dipole. The plus side of one dipole is attracted towards the minus side of another dipole, and vice versa: the molecules attract each other.

The physics behind the van der Waals force: neutral atoms or molecules electrically polarize each other. From a lecture course at Columbia University.

However, the dipole of one molecule does only form in the presence of another particle. On its own, the molecule relaxes to an electrical equilibrium state. So, when two molecules meet, which one will form and stretch out the plus side to the other and which one the minus? The answer is very strange: each molecule will form a plus and a minus pole at the same time; the pole will be in a quantum-superposition state of plus and minus.

According to quantum physics [4], the world is teeming with possibilities, virtual processes where Nature tries out infinitely many things at the same time, before some of them materialize into solid fact. Sometimes they never do, but the virtual processes may still have a real effect. The van der Waals force is a good example: it is not necessary that the molecules decide which one points the plus side and which one the minus side to each other; they attract each other regardless.

Casimir cavity: the left picture shows a cavity made by two metal plates (by two mirrors); the right picture illustrates standing electromagnetic waves in the cavity. Even if the cavity is empty, without any electromagnetric field inside, the sheer possibility that such standing waves may exist is important in the Casimir effect.

Imagine that you replace the molecules by larger bodies, say glass or metal plates. Even if the plates are electrically neutral in total, virtual patterns of charge variations could form on the surfaces, local pluses on one plate that are attracted to minuses on the other plate, and vice versa. Like in the case of the van der Waals force between molecules, the pluses and minuses are undecided, they are in quantum-superposition states, even across relatively large distances (a few 100 nanometers) and between extended bodies; causing a force known as the Casimir force.

Casimir effect and vacuum fluctuations: roughly speaking, the difference in the pressure of the quantum vacuum inside and outside the cavity causes the plates to attract each other.

Maritime analogy of the Casimir force [5]. In calm weather and without any water currents, ripples on the sea may cause two tall ships to attract each other, with potentially catastrophic consequences. The pressure difference of the ripples between the ships and the ripples outside them causes the attractive force, similar to the Casimir force. In the Casimir effect, the ships are the cavity plates and the quantum ripples of the empty electromagnetic field play the role of the water waves.

Hendrik Casimir discovered the theoretical possibility of such a force in 1948 [6]. He told Niels Bohr about his strange and surprisingly simple formulas during a walk [4]. Bohr suggested in a cryptic remark that one can also understand the force between the plates as being caused by the zero-point energy of the electromagnetic field, by vacuum fluctuations [4]. Empty space is not empty, but is filled with the quantum vacuum, with endless virtual processes. The energy of the quantum vacuum, the zero-point energy is infinite according to our present theories. Clearly, this infinity is an artifact - it would make the electromagnetic field infinitely massive, because energy and mass are related according to Einstein's E=mc^2. The empty electromagnetic field would collapse under the weight of its own gravity. Some unknown mechanism beyond quantum electromagnetism must regularize the infinity of the electromagnetic vacuum energy. Nevertheless, the zero-point energy results in perfectly finite and experimentally confirmed facts, for example the Casimir force.

Apparatus for measuring and manipulating the Casimir force. Instead of two parallel plates, a Gold sphere and a nano-fabricated silicon swing form a cavity. The torsion of the swing measures the Casimir force. These experiments are done in the group of Federico Capasso.

In 1997 the first precise observation of the Casimir force was reported [7]. Since then, a series of ever-more sophisticated experiments showed that the Casimir force is not only real and does agree with quantum theory to an astonishing accuracy, but that it can be applied in nano-fabricated devices such as Microelectromechanical Systems (MEMS) [8,9]. MEMS combine tiny mechanical structures with electronics on one chip. For example, the chip that triggers the airbag in a car contains both the mechanical elements for measuring violent de-acceleration and the electronics needed for deciding when to explode the airbag.

Accelerometer and electronics on one chip - the trigger chip of an airbag, for example. From Sandia's MEMS page.

The Casimir force is the ultimate cause of friction in the nano-world. Micro- or nano-machines could run smoother and with less or no friction at all if one can manipulate the Casimir force.

Micro-machinery. From Sandia's MEMS page.

Imagine that you put a transparent material between the Casimir plates. The material may influence the way in which the virtual dipoles of the plates respond to each other, or, equivalently, the distribution of the zero-point energy. We found out that the plates repel each other if the material is electromagnetically left-handed [1]. Such materials show negative refraction.

Negative refraction. (a) shows an empty glass, (b) a glass filled with an ordinary medium with positive refractive index, such as water; the straw inside the glass is refracted. (c) shows what would happen if the water is replaced by a negatively refracting medium. From the Nanophotonics group at the Karlsruhe Institute of Technology.

Left-handed (or negatively-refracting) materials turn out to transform space for electromagnetic fields and their vacuum fluctuations [2,10].

A negatively refracting medium transforms space [2,10]. The top figure shows the graph of a coordinate transformation from the real Cartesian x to x'. The medium shown in the lower picture turns out to perform this transformation. The transformation changes right-handed into left-handed coordinates and so the medium creates left-handed electromagnetism. Our picture shows why left-handed media make perfect lenses [11]: each point x in physical space corresponds to one x', but this x' has two more images in x, one inside the device and one outside. Since this map is perfect in principle, the electromagnetic fields at the three x points are identical; the device acts as a perfect lens. The lower picture shows that light rays are negatively refracted. Such transformations turn the attractive Casimir force in x' space into a repulsive force in real space.

In transformed space the Casimir plates attract each other, but the transformation causes the plates to repel each other in real space. One plate could hover over the other at the distance where the repulsive Casimir force of the quantum vacuum balances the weight of the plate; the plate levitates on, literally, nothing [1].

Levitating mirror [1].

Our idea [1] is not the only option of making the Casimir force repulsive [12], but it may shed light on the general mechanism acting behind the scenes, because our theory is inspired by a simple picture of how space is transformed; it visualizes Casimir repulsion. As you have seen, Quantum Levitation uses a fascinating piece of quantum physics and it may find applications in nanotechnology. Incredible!

References

1. U. Leonhardt and T.G. Philbin, Quantum levitation by left-handed metamaterials, New Journal of Physics 9, 254 (2007).
2. U. Leonhardt and T.G. Philbin, Quantum optics of spatial transformation media, Journal of Optics A (in press).
3. K. Autumn, M. Sitti, Y.A. Liang, A.M. Peattie, W.R. Hansen, S. Sponberg, T.W. Kenny, R. Fearing, J. N. Israelachvili, and R. J. Full, Evidence for van der Waals adhesion in gecko setae, Proceedings of the National Academy of Sciences 99, 12252 (2002).
4. P. W. Milonni, The Quantum Vacuum (Academic, London, 1994).
5. S.L. Boersma, A maritime analogy of the Casimir effect, American Journal of Physics 64, 539 (1996).
6. H. Casimir, On the attraction between two perfectly conducting plates, Proceedings of the Royal Netherlands Academy of Arts and Sciences, B51, 793 (1948).
7. S.K. Lamoreaux, Demonstration of the Casimir Force in the 0.6 to 6 µm Range, Physical Review Letters 78, 5 (1997).
8. H.B. Chan, V.A. Aksyuk, R.N. Kleiman, D.J. Bishop, and F. Capasso, Quantum Mechanical Actuation of Microelectromechanical Systems by the Casimir Force, Science 291, 1941 (2001).
9. Ph. Ball, Feel the force Nature 447, 772 (2007).
10. U. Leonhardt and T.G. Philbin, General relativity in electrical engineering, New Journal of Physics 8, 247 (2006).
11. J. B. Pendry, Negative Refraction Makes a Perfect Lens, Physical Review Letters 85, 3966 (2000).
12. See e.g. E. Buks and M. L. Roukes, Quantum physics: Casimir force changes sign, Nature 419, 119 (2002).

About the Author

Prof. Ulf Leonhardt is Chair in Theoretical Physics of the University of St Andrews. His research interests go to  Fibre-optical horizons, Quantum levitation, Invisibility, Quantum catastrophes and Light in moving media.  He is the author of the book “Measuring the Quantum State of Light” , book contributions and numerous scientific articles. He is also an able and intriguing science writer.