Mesoscopic structure formation in condensed matter due to vacuum fluctuations

An observable influence of zero-point fluctuations of the vacuum electromagnetic field on bound electrons is well known in the hydrogen atom, where it produces the Lamb shift. Here, we adapt an approach used to explain the Lamb shift in terms of a slight expansion of the orbits due to interaction with the zero-point field and apply it to assemblies of $N$ electrons that are modeled as independent atomically bound two-level systems. The effect is to stabilize a collective ground-state energy, which leads to a prediction of novel effects at room temperature for quasi-two-dimensional systems over a range of parameters in the model, namely, $N$, the two-level excitation energy $\hslash \omega$ and the ionization energy $\hslash \omega +\varepsilon$. Some mesoscopic systems where these effects may be observable include water sheaths on protein or DNA, surfaces of gaseous nanobubbles, and the magnetic response of inhomogeneous, electronically dilute oxides. No such effects are envisaged for uniform three-dimensional systems.

Figure 1

(a) Energy levels of a bound electron; $|0\rangle$ and $|1\rangle$ are the ground and excited states, $(\hslash \omega +\varepsilon )$ is the ionization energy. (b) Energy levels of each of the $N$ electrons that interact with the vacuum electromagnetic field. The collective ground-state energy is lowered by $G^2\hslash \omega$, where $G^2\propto N$, but also $G^2\ll 1$.

Figure 2

Schematic illustration of the systems considered: (a) is a 3D system, where the $N$ atoms each with a bound electron- are distributed throughout the bulk; (b) is a quasi-2D system, where the $N$ atoms are found only in a surface layer. In each case illustrated, half the available atoms have a bound electron.

Figure 3

Variation with photon wavelength $\lambda$ of the value of $G$ corresponding to stabilization of the mesoscopic ground state at room temperature and the numbers $N$ of bulk and surface atoms as a function of $\lambda$ (red and blue solid lines). The dashed lines are the numbers of atoms required for mesoscopic structure formation at room temperature. Note that stability is only possible in 2D, in the blue-shaded area of the diagram, where $N^{\mathrm{surf}}>N_2^{\mathrm{RT}}$.