Topological Stability of Stored Optical Vortices

We report an experiment in which an optical vortex is stored in a vapor of Rb atoms. Because of its $2\pi$ phase twist, this mode, also known as the Laguerre-Gauss mode, is topologically stable and cannot unwind even under conditions of strong diffusion. For comparison, we stored a Gaussian beam with a dark center and a uniform phase. Contrary to the optical vortex, which stays stable for over $100\mu \mathrm{s}$, the dark center in the retrieved flat-phased image was filled with light after a storage time as short as $10\mu \mathrm{s}$. The experiment proves that higher electromagnetic modes can be converted into atomic coherences and that modes with phase singularities are robust to decoherence effects such as diffusion. This opens the possibility to more elaborate schemes for classical and quantum information storage in atomic vapors.

Figure 1

Experimental setup for optical vortex storing. (a) The energy level scheme of the $D1$ transition of ${}^{87}\mathrm{Rb}$ showing the three levels of the $\Lambda$ system and the pump and probe transitions. (b) The experimental setup. ECDL—external-cavity diode laser, PBS—polarizing beam splitter, AOM—acousto-optic modulator, $\lambda /2$—half wave plate, $\lambda /4$—quarter wave plate, MS—magnetic shields, HC—Helmholtz coils.

Figure 2

The effect of diffusion on a helical beam and a flat phase beam. The left column shows the initial shape of a helical (top) and a flat phase (bottom) beam before any diffusion (off-resonance). The right column shows both beams after storage of $30\mu \mathrm{s}$. The center remains dark in the helical beam while completely filling with light in the flat phase beam.

Figure 3

Averaged radial cross sections of the restored helical beam for different storage durations (30, 70, and $110\mu \mathrm{s}$). The cross sections of the off-resonance and slowing cases are also shown (the small difference between them is mainly due to diffusion during the slowing). It is evident that the center remains completely dark, even for the longest storage duration. The theoretical predictions (solid lines) are in excellent agreement with the experimental results (symbols). As explained in the text, the effect of diffusion on this transverse mode is to increase its waist $w_0$ to $\sqrt{w_0^2+4Dt}$, where $D=11{\mathrm{cm}}^2/\mathrm{s}$.

Figure 4

Averaged radial cross sections of the restored flat phase beam with a dark center for two storage durations (10 and $30\mu \mathrm{s}$). Also shown are the slowed and off-resonance cross sections. Due to the diffusion, the dark center is already filled when the beam is only slowed. After a storage time of $30\mu \mathrm{s}$, the dark center was completely filled with light. The theoretical prediction (solid lines) captures this essential phenomenon but is less accurate than for the helical case. We attribute this discrepancy mostly to the more complex propagation of this imaged beam, which, contrary to the helical case, is not a transverse mode of the electromagnetic field.