Observation of Nonspreading Wave Packets in an Imaginary Potential

We propose and experimentally demonstrate a method to prepare a nonspreading atomic wave packet. Our technique relies on a spatially modulated absorption constantly chiseling away from an initially broad de Broglie wave. The resulting contraction is balanced by dispersion due to Heisenberg’s uncertainty principle. This quantum evolution results in the formation of a nonspreading wave packet of Gaussian form with a spatially quadratic phase. Experimentally, we confirm these predictions by observing the evolution of the momentum distribution. Moreover, by employing interferometric techniques, we measure the predicted quadratic phase across the wave packet. Nonspreading wave packets of this kind also exist in two space dimensions and we can control their amplitude and phase using optical elements.

Figure 1

Formation of a nonspreading Michelangelo wave packet for the center-of-mass motion of an open two-level atom (a). The resonant interaction with a standing light wave (b) leads to an array of harmonic imaginary potentials. The normalized diffraction efficiencies derived from the momentum distributions (inset) approach a steady state as a function of the interaction length $\Delta z$ demonstrating the successful realization of stationary wave packets. The solid curves result from a numerical integration of the Schrödinger equation [17] with the Rabi frequency ${\Omega }_0=0.4\Gamma$. The dashed lines correspond to the Raman-Nath approximation, revealing that the interplay between absorption and quantum spreading is essential for obtaining a steady state.

Figure 2

Experimental verification of the scaling law $t_0\sim 1/{\Omega }_0$ connecting the characteristic time $t_0\equiv z_0/v$ when Michelangelo wave packets form and the Rabi frequency ${\Omega }_0$. The line is a guide to the eye. We measure the zeroth order diffraction efficiency as a function of $\Delta z$ (inset) for different Rabi frequencies. The crossing point between the linear extrapolation of the short and long-time limits yields $z_0$.

Figure 3

Measurement of the phase of a Michelangelo wave packet using an interferometric setup (a) consisting of the absorptive and probing standing waves. The inset in (b) shows typical interference patterns for different output directions obtained by scanning the relative position of the second (thin) standing light wave for a given interaction length $\Delta z$. For large values of $\Delta z$ the phase shifts (b) of the different interferometer outputs relative to the zeroth order level off, indicating stationary phases of the wave packet.

Figure 4

Probability density $|\phi (x){|}^2$ (solid line) and absolute value of phase $|\varphi (x)|$ (dotted line) of Michelangelo wave packets in the potential $U_{2n}$.