Disorder in Quantum Vacuum: Casimir-Induced Localization of Matter Waves

Disordered geometrical boundaries such as rough surfaces induce important modifications to the mode spectrum of the electromagnetic quantum vacuum. In analogy to Anderson localization of waves induced by a random potential, here we show that the Casimir-Polder interaction between a cold atomic sample and a rough surface also produces localization phenomena. These effects, that represent a macroscopic manifestation of disorder in quantum vacuum, should be observable with Bose-Einstein condensates expanding in proximity of rough surfaces.

Figure 1

(a) Cigar-shaped BEC of width $\sigma$ and axial size $2L_{\mathrm{TF}}$ above a rough surface. Here $L_{\mathrm{TF}}$ is the initial Thomas-Fermi length of the BEC. Dimensionless response functions for a perfectly reflecting surface: (b) $g^{(1)}(\mathcal{Z})$ ($\mathcal{Z}=kz$) and (c) $g^{(2)}({\mathcal{Z}}_i,{\mathcal{Z}}_j)$ (${\mathcal{Z}}_i=k_{i}z$ and ${\mathcal{Z}}_j=k_{j}z$).

Figure 2

$\gamma (k)$ vs $k\xi$ from Eq. (6). The atom-surface distance is $z_0=1.5\mu \mathrm{m}$. Inset: The same data in log-lin scale, the maximum length scale to be measured $L_{\max }=1\mathrm{mm}$, and the value $k^{*}={\gamma }^{-1}(1/L_{\max })$ separating localized (Loc.) from delocalized (Deloc.) modes.

Figure 3

BEC density (arbitrary units) vs position. Both the perturbative theory described by Eqs. (4, 5, 6) (solid line) and the full numerical simulation (dots) are computed by using the first-order approximation for the CP potential at $z_0=1.5\mu \mathrm{m}$. The surface profile is averaged over 40 realizations. Time corresponds to ${\omega }_{x}t=28$.

Figure 4

BEC density (arbitrary units) vs position, after ${\omega }_{x}t=14$. The profile is averaged over 40 realizations at a distance of $z_0=1.0\mu \mathrm{m}$. Numerical simulation (dots) includes both the first- and second-order terms of the lateral CP potential $U_L(x,z)$. The wing is fitted by a power law $\overline{n(x)}\propto 1/x^{\nu }$ with $\nu =1.84$ (solid line). Inset: Zoom of the numerical data in log-lin scale.