Historical research guide · 0477 · 2016
Damped vacuum states of light
T G Philbin
In one minute
Most Casimir calculations only ever ask for one number — the total energy, or the force it produces — and that habit hides the structure underneath. Thomas Philbin of the University of Exeter opens it up. He takes the simplest interesting case, a slab of real material with light running straight through it, and keeps everything the usual shortcuts throw away: the material absorbs, it disperses, and it obeys the Kramers-Kronig relations that any real substance obeys. Then he asks what the vacuum energy is at each separate frequency. The answer is that the slab pushes some frequencies down below their empty-space value and pushes others up. For a gold-like metal, everything below the plasma frequency is damped and everything above it is lifted. Each of those shifts is finite on its own, with no regularisation needed. And the grand total, once regularised the standard way, still comes out positive.
Connection
Chapter 2 says the vacuum is a structured medium whose spectrum you can reshape with ordinary matter, and this is that statement made quantitative frequency by frequency rather than lumped into a single force. It also settles a point of language the site cares about: the modes that fall below the free-space level are not holes in nothing, they are ordinary modes sitting below the ambient level, and the total stays positive. Chapter 5 gets the bridge Philbin draws himself — this is the physics of a quantum damped oscillator, the same object nanomechanics works with. Read it beside the same author’s fibre-optical analogue of an event horizon at /library/stm-a5c8b5b2b0.
From the source
One-dimensional propagation of quantum light past a block of material can be treated exactly, with full account of dispersion and absorption, by writing macroscopic electromagnetism as a closed system of fields coupled to a reservoir and diagonalising the Hamiltonian — no idealised boundary conditions and no dielectric functions that violate the Kramers-Kronig relations.Section 2, ‘Set-up’; Eqs. 1 to 3; Figure 1
The zero-point electric and magnetic field uncertainties outside the block oscillate with distance from its face, yet the energy per unit length outside is exactly the free-space value: the material dependence cancels for modes propagating perpendicular to the boundaries, so the whole change in zero-point energy sits inside the block.Section 3, Eqs. 11 and 12; Section 4, Eq. 22
Inside the block, the zero-point energy per unit frequency is damped below the free-space value for some frequencies and raised above it for others — for a Drude model of gold with plasma frequency 8.45 eV and damping 0.047 eV, the energy is suppressed everywhere below the plasma frequency and enhanced above it, for block lengths of both 1 µm and 10 µm.Section 5, ‘Examples’; Eq. 28; Figure 2
The change in zero-point energy caused by the block is finite at every single frequency, before and after regularisation; only the sum over all frequencies diverges, so the frequency-resolved result does not depend on the choice of regulariser.Section 4, paragraph following Eq. 24
After the standard Casimir regularisation — the same one used to predict measured Casimir forces — the Casimir energy per unit frequency oscillates between positive and negative values, so individual modes sit below the free-space level, while the total Casimir energy comes out positive for every block length and for both metals and non-metallic dielectrics.Section 4, Eqs. 26 and 27; Figures 3 and 5; Section 6, Conclusions
Because the quantum theory of macroscopic electromagnetism is a theory of quantum damped harmonic oscillators, the same modifications of zero-point energy should appear in nanomechanical systems, where a single oscillator is coupled to a reservoir instead of the field’s infinite set.Section 1, Introduction; Section 6, Conclusions
The way in
https://doi.org/10.1088/2040-8978/18/9/095201How to cite it
T G Philbin (2016) Damped vacuum states of light doi:10.1088/2040-8978/18/9/095201