A Maxwell Fish-Eye Lens in a Bose-Einstein Condensate

Abstract

We experimentally realize an analogue of the optical Maxwell fish-eye lens (MFEL) using phononic excitations in a Bose-Einstein condensate (BEC). A MFEL is characterized by a radially symmetric, spatially varying refractive index with the remarkable property that rays emitted from any point within the lens are perfectly focused at their image points. While the implementation of such gradient-index lenses is challenging in conventional optical systems, BECs offer a highly tunable platform in which the spatially varying speed of sound of collective excitations -- phonons, the acoustic-wave analogues of photons -- can be engineered and their dynamics observed in real time. Time-resolved measurements of phonon wavefronts reveal focusing behavior that shows good agreement with analytical theory and numerical simulations. This work provides both a geometric and physical framework for engineering effective refractive indices using ultracold atoms, and simulating wave propagation on effective spherical geometries.

Figures (3)

• Figure 1: a. Schematic diagram of the refractive index for a 2D Maxwell fish-eye lens, showing light rays emanating from a particular point and perfect focusing at the corresponding image point. The solid and dashed black lines respectively represent the trajectories with and without the presence of a reflecting mirror at $r=R$ (solid gray line). b. Corresponding experimental condensate atomic density profile. The white (experiment) and yellow (theory) lines are 1D slices, and separate a 3D from a 2D depiction of the density. c. Light rays (great circles) on the surface of a sphere with constant refractive index $n_{1}$, whose stereographic projection onto the 2D equatorial plane corresponds to the ray trajectories in the 2D lens. The equator plays the role of the mirror in this representation.
• Figure 2: Snapshots in time of our effective Maxwell fish-eye lens focusing dynamics: analytic wave-fronts represented on the surface of a sphere (first row), idealized Gross-Pitaevskii equation simulation (second row), experimental results (third row), and again simulation but now using the initial $t=0$ experimental background density (fourth row). The left column shows the initial density distribution of the lens. The middle columns show the density difference for different times, $\delta\rho(\mathbf{r},t)=\rho(\mathbf{r},t)-\rho_{BG}(\mathbf{r})$, where $\rho_{BG}(\mathbf{r})$ is the initial density without a perturbation, and the right column shows the fidelity (\ref{['eq:fidelity']}) between the density difference at a given time $\delta \rho(\mathbf{r},t)$ versus the expected density difference if perfect focusing were to occur at the antipodal point, $\delta\rho(-\mathbf{r},0)$. The maximum overlap is at $t = 31.5\,\mathrm{ms}$ (vertical dashed line), in agreement with the predicted time of $T = 31.5\,\mathrm{ms}$ obtained via equation (\ref{['eq:total_time']}) with a measured speed of sound in the center of $c_0\approx1.8\,\mathrm{\mu m/ms}$ and $R=36$$\mu$m.
• Figure 3: Focusing fidelity for different initial wavepacket positions $r_{0}$: the density at $t = 0\,\mathrm{ms}$ including the wavepacket indent (first row), the density difference at the analytically predicted focusing time of $T = 31.5\,\mathrm{ms}$ (second row), and fidelity (\ref{['eq:fidelity']}) over time for all initial positions (rightmost image). The fidelity shows a clear peak around $t = 31.5\,\mathrm{ms}$ (vertical dashed line). For larger initial displacements from the central position the peak shifts slightly towards later times.