Observation of Parity-Time Symmetry in Optically Induced Atomic Lattices

We experimentally demonstrate $PT$-symmetric optical lattices with periodical gain and loss profiles in a coherently prepared four-level $N$-type atomic system. By appropriately tuning the pertinent atomic parameters, the onset of $PT$-symmetry breaking is observed through measuring an abrupt phase-shift jump between adjacent gain and loss waveguides. The experimental realization of such a readily reconfigurable and effectively controllable $PT$-symmetric waveguide array structure sets a new stage for further exploiting and better understanding the peculiar physical properties of these non-Hermitian systems in atomic settings.

Figure 1

(a) Experimental setup. ${\mathit{E}}_p^{\prime }$ and ${\mathit{E}}_p$ are pump beams from the same laser, and so are the coupling beams ${\mathit{E}}_c^{\prime }$ and ${\mathit{E}}_c$. The reference beam intersects with ${\mathit{E}}_s$ to generate the reference interference. (b) The energy-level diagram of the four-level $N$-type configuration in ${}^{85}\mathrm{R}\mathrm{b}$ atomic vapor. (c) The spatial arrangements of the signal field, coupling-field lattice, and pump-field lattice. $x$ and $z$ represent the transverse and longitudinal directions of beam propagation, respectively. $\mathrm{\Delta }d$ marks the displacement between the two lattices.

Figure 2

The real (dispersion) and imaginary (gain or absorption) parts of the susceptibility versus ${\mathrm{\Delta }}_s$ at (a) ${\mathrm{\Omega }}_p=0$ and (b) ${\mathrm{\Omega }}_p\ne 0$, respectively. The (c) real and (d) imaginary parts for periodic lattices of the refractive index as a function of position $x$ with the coupling intensities spatially modified. ${\mathrm{\Omega }}_s=2\pi \times 10\mathrm{MHz}$, ${\mathrm{\Omega }}_c=2\pi \times 0.2[1+\cos (\pi x/2)]\mathrm{MHz}$, ${\mathrm{\Omega }}_p/2\pi =6\mathrm{MHz}$, ${\mathrm{\Delta }}_p=40\mathrm{MHz}$, ${\mathrm{\Delta }}_c=-100\mathrm{MHz}$, and ${\mathrm{\Delta }}_s\approx -2\pi \times 15\mathrm{MHz}$.

Figure 3

(a) Image and intensity profile of ${\mathit{E}}_s$ without interacting with atoms. (b) Signal beam after propagating through the coupling lattice. (c) Simultaneous gain and loss profiles on ${\mathit{E}}_s$ with both lattices turned on. (d) Dependence of the gain-to-loss ratio on ${\mathrm{\Delta }}_p$. The squares are experimental observations, and the solid curve is the theoretical prediction. The observed gain-loss profiles are presented at (d1) ${\mathrm{\Delta }}_p=30\mathrm{MHz}$ and (d2) 10 MHz.

Figure 4

Selected gain and loss channels for measurements are marked with $G$ and $L$, respectively. (a) Reference interference fringes generated by the reference beam and the signal beam without diffraction. The two solid lines mark the centers of two adjacent fringes. (b)-(f) Observed phase differences (marked by the pair of one-way arrows) between the adjacent gain and loss channels with ${\gamma }_G/{\gamma }_L$ being 0, 0.4, 0.6, 0.8, and 1.0, respectively. (g) Measured dependence of phase difference on ${\gamma }_G/{\gamma }_L$. The left (gray) and right (green) parts are the regions below and above the $PT$-symmetry breaking threshold, respectively. (h) Theoretical simulations of phase difference according to the coupled equations for ten waveguides at a propagation distance of $z=10$. The breaking threshold (the dotted curve) decreases with an increasing gain-to-loss ratio. The dimensionless $z$ is scaled by the Rayleigh range $k{x}_0^2$ ($k=2\pi /\lambda$, and $x_0$ is the waveguide width).