Historical research guide · 1063 · 2002
Focusing vacuum fluctuations. II
L. H. Ford · N. F. Svaiter
In one minute
Larry Ford and Nami Svaiter point a mirror at empty space. Not at a light source — at the vacuum, whose fields never stop fluctuating. Their answer, worked out by applying ordinary ray-tracing optics to quantum fields, is that a parabolic mirror focuses vacuum fluctuations the way it would focus light: the mean squared electric field near the focus grows as an inverse power of the distance from the focal line, and it does so because the reflected rays interfere with each other. This second paper extends the first from points on the symmetry axis to any direction from the focus, and corrects an error in it. The striking result is the sign. Depending on how wide the mirror is, and which way you approach the focus, the mean squared electric field comes out negative or positive. Negative means a repulsive force on a nearby atom and a column of energy density sitting below the ambient vacuum level. Positive means an attractive force, and the possibility of trapping an atom with no applied field at all.
Connection
Chapter 2 argues that the vacuum is a medium you can shape, and this is a mirror shaping it — geometry alone deciding whether the field near a focus sits above or below the ambient level. Chapter 6 gets the experiments: Ford and Svaiter name three, and the boldest of them, holding a cold atom in place with nothing but a focused vacuum and no applied electromagnetic field, would be the plainest demonstration yet that empty space has a structure you can engineer.
From the source
A parabolic mirror focuses vacuum fluctuations. Ford and Svaiter show that near the focus the mean squared scalar field and the mean squared electric field grow as inverse powers of the distance from the focus, rather than as inverse powers of the distance from the mirror surface, and that the enhancement comes from the interference between different reflected rays rather than between the incident and reflected ray.Section I, and Section II equations 4 to 6 (arXiv:quant-ph/0204126v2)
Geometry alone sets the sign. For a parabolic cylinder whose half-angle is less than a right angle, the mean squared electric field is negative everywhere near the focal line and an atom feels a force pushing it away. For a wider mirror, between a right angle and 120 degrees, the sign depends on the direction of approach — nearly perpendicular to the symmetry axis the mean squared electric field turns positive and the force becomes attractive.Section V, equations 34 to 36 with Figure 4, and Section IX (arXiv:quant-ph/0204126v2)
Where the mean squared electric field is negative there are columns of energy density below the ambient vacuum level running parallel to the focal line of the mirror. In the geometric optics approximation the mean squared electric and magnetic fields are equal, so the local energy density is just the mean squared electric field — which makes the shaped vacuum, not a source, the thing that sets it.Section IX, Discussion and Conclusions (arXiv:quant-ph/0204126v2)
The effect has a named laboratory test with numbers attached. Send well-collimated sodium atoms parallel to the focal line and measure their deflection: the predicted fractional deflection is of order a quarter for a passage about one micrometre from the focus over a millisecond, which is the same style of measurement Sukenik and colleagues already used to confirm the Casimir and Polder prediction for a flat plate. An atom interferometer with one arm near the focal line offers a second route, with a predicted phase difference of about 0.04 radians against an instrument sensitivity of order one ten-thousandth of a radian.Section VIII, equations 62 to 65 (arXiv:quant-ph/0204126v2)
The boldest proposal is trapping. A mirror wider than a right angle creates a region where the mean squared electric field is positive, and atoms cooled to about two nanokelvin could be held within roughly a micrometre of the focus. Ford and Svaiter stress what would make this different from every atom trap in use: it would require no applied classical electromagnetic fields at all.Section VIII, equation 66, and Figure 13 with the potential-minimum argument (arXiv:quant-ph/0204126v2)
The authors state the limits of their own model. The divergences that appear at particular geometries come from two idealisations — a mirror that reflects perfectly at every frequency and has perfectly sharp edges. A real mirror is a good reflector only for wavelengths longer than the metal’s plasma wavelength, about 84 nanometres for aluminium, and surface roughness and the atomic-scale breakdown of a continuous surface are needed as well to keep the mean squared field finite. With any sufficiently sharp short-wavelength cutoff the results are bounded by the inverse square and inverse fourth power of that cutoff wavelength.Section VII, equations 60 and 61, with the diffraction estimate in the Appendix (arXiv:quant-ph/0204126v2)
The way in
https://doi.org/10.1103/physreva.66.062106How to cite it
L. H. Ford, N. F. Svaiter (2002) Focusing vacuum fluctuations. II doi:10.1103/physreva.66.062106