Negative refraction for incoherent atomic matter waves

In a recent paper [Phys. Rev. Lett. 102, 140403 (2009)], Baudon et al. reported that a comoving potential acting on atoms could mimic negative-index-refraction properties of left-handed photonic metamaterials. We show in this article that a better approximation is necessary in order to describe the real physical system proposed by Baudon et al. We derive analytical formulas and compare their solutions with numerical simulations. It turns out that instead of negative refraction, it is simply atomic reflection that occurs in the above-mentioned comoving potential. Furthermore, we find that the wave packet of the reflected atomic beam by comoving potentials can be narrowed by negative dispersion. Finally, we identify a potential configuration, near the interface with a cubic-type potential barrier, that may lead to the observation of negative refraction with incoherent matter waves.

Figure 1

Analytical result of the relative position of the wave-packet center $\Delta u_a$ from Eq. (19) (dashed curve) in comparison to the numerical result $\Delta u_n$ (solid curve) and previous analytical solution from Eq. (2) (dotted-dashed curve) for $\alpha =72$ and $\epsilon =-0.05$ (see text).

Figure 2

Absolute position of the wave-packet center $u\left(\tau \right)$ in units of $\lambda /2\pi$ as a function of time, calculated under different conditions: quantum numerical simulation in presence of group velocity inversion $(\epsilon =-1.25$ and $\alpha =37$ and $\delta k=\frac{\kappa }{2\pi 0.1})$ (bold solid curve); classical numerical simulation with similar potential (dashed curve); harmonic potential approximation for deep potential (thin solid curve); shallow potential approximate result from Eq. (13) for an identical potential (dotted-dashed curve); free propagation case (dashed line).

Figure 3

Wave packet versus position $x$ at different times during the interaction with a periodical potential, whose parameters are the ones given in Fig. 2. (Solid line) initial wave packet with center position $x_M\left(0\right)=\lambda$; (dashed line) intermediate-time wave packet, after reflection; (dotted-dashed line) wave packet with complete inversion of the group velocity.

Figure 4

Numerical calculation of the group velocity (dashed line) and phase velocity (solid line) as a function of interacting time for a spatially narrow wave packet ${\delta }_x=0.02\lambda$, with potential depth $\epsilon =-1.0$ and $\alpha =36$. We added an arbitrary reference energy $E_0=300\hslash {\omega }_{\kappa }$ for the calculation of the phase velocity, which ensures phase velocity is always positive before the turning point but does not change the interpretation of the result.

Figure 5

Center position of the wave packet with respect to its initial position (solid line) and wave-packet dispersion (dashed line). Parameters of the simulation are the ones of Fig. 2, except $\epsilon =-1.0$ and $x_0=0$. Maximum inverted velocity (slope of the center position curve) is reached at $t\sim 0.07/{\omega }_{\kappa }$. Large wiggles appear for $t>0.08/{\omega }_{\kappa }$.

Figure 6

Numerical calculation of the group velocity (dashed line) and phase gradient at wave-packet center (solid line) as a function of interacting time for a spatially narrow wave packet ${\delta }_x=0.02\lambda$ launched onto a cubic-type potential barrier $V\left(x\right)\propto x+a_3{x}^3$ where $a_3=\frac{1}{6}$ (see text).

Figure 7

Probability densities (thin curves) and phases (bold curves) of the wave packets just before phase gradient inversion at $t=0.048{\omega }_{\kappa }^{-1}$ (dashed curves) and close to group velocity inversion at $t=0.06{\omega }_{\kappa }^{-1}$ (solid curves), with computation parameters as given in Fig. 6.