Cloaking of Matter Waves

Invariant transformation for quantum mechanical systems is proposed. A cloaking of matter wave can be realized at given energy by designing the potential and effective mass of the matter waves in the cloaking region. The general conditions required for such a cloaking are determined and confirmed by both the wave and particle (classical) approaches. We show that it may be possible to construct such a cloaking system for cold atoms using optical lattices.

Figure 1

Schematic of a spherical cloaking system with interaction potential and effective mass tuned in the shell region between $r_1$ and $r_2$. An impinging particle travels along $+y$ direction and has an initial momentum $P_0$, and impact parameter $b$. After passing throughout the system it leaves at an angle $\alpha$.

Figure 2

The trajectories of particles passing through a cloaking system with $r_2=2r_1$ for (a) $\Delta E=0$ (b) $\Delta E=-0.1E_0$, and (b) $\Delta E=0.1E_0$.

Figure 3

(a) The dependence of effective mass and band edge energy on the amplitude of the optical lattice potential $s$ (b) The plot of the radial scaling function $g$ (black), effective mass along radial direction $m_r$ (gray dashed) and potential $V$ (gray) for a reduced cloaking design with $r_2=4r_1$, $\overline{m}=10m_0$ and $\delta =r_1$. (c) The combined optical and magnetic potential profile to achieve the proposed cloaking system for $r_1=100\mu \mathrm{m}$, $r_2=400\mu \mathrm{m}$ and $\delta =100\mu \mathrm{m}$. Inset is a magnified view of the shadowed area.