Anti-Parity-Time Symmetric Optical Four-Wave Mixing in Cold Atoms

Non-Hermitian optical systems with parity-time ($PT$) symmetry have recently revealed many intriguing prospects that outperform conservative structures. The previous works are mostly rooted in complex arrangements with controlled gain-loss interplay. Here, we demonstrate anti-$PT$ symmetry inherent in the nonlinear optical interaction based upon forward optical four-wave mixing in a laser-cooled atomic ensemble with negligible linear gain and loss. We observe that the pair of frequency modes undergo a nontrivial anti-$PT$ phase transition between coherent power oscillation and optical parametric amplification in presence of a large phase mismatch.

Figure 1

Anti-$PT$-symmetric forward four-wave mixing (FFWM) in a two-dimensional (2D) magneto-optical trap (MOT) of ${}^{85}\mathrm{Rb}$ atoms. (a) ${}^{85}\mathrm{Rb}$ atomic energy-level diagram: $|1⟩=|5S_{1/2},F=2⟩$, $|2⟩=|5S_{1/2},F=3⟩$, $|3⟩=|5P_{1/2},F=3⟩$, and $|4⟩=|5P_{3/2},F=3⟩$. (b) Geometrical arrangement on four interacting fields inside the MOT, where both pump ($E_p$) and coupling ($E_c$) beams of the same diameter of 1.54 mm are twisted by an angle of $\theta =0.2°$ to $+z$ axis. The Stokes ($E_s$) and anti-Stokes ($E_{as}$) light collinearly propagate along the $+z$ axis with the same waist diameter of $300\mu \mathrm{m}$ at the MOT center. (c) Schematic of the experimental setup. In the injection part, the coupling (${\omega }_c$) and anti-Stokes (${\omega }_{as}$) beams, same to the pump (${\omega }_p$) and Stokes (${\omega }_s$) beams, are first combined by a beam splitter (BS) and polarization beam splitter (PBS) and then directed through a quarter-wave plate to generate the desired circular polarizations. In the collection part, the output Stokes and anti-Stokes beams are first separated through their polarizations and then measured by photomultiplier tubes (${\mathrm{PMT}}_s$ and ${\mathrm{PMT}}_{as}$) after the narrow band Fabry-Perot (FP) filters.

Figure 2

Comparison of phase mismatch $\mathrm{\Delta }k$, nonlinear coupling coefficient $\kappa$, linear Raman gain coefficient $g$ of the Stokes field, and linear absorption coefficient $\alpha$ of the anti-Stokes field as functions of atomic density $N$ or $\delta$ at the two-photon resonance. The shaded color areas mark two different phases of anti-$PT$ symmetry.

Figure 3

Transmission spectra of Stokes (blue) and anti-Stokes (orange) outputs at different $\delta$ for seeding only anti-Stokes (a), and only Stokes (b). The theoretical simulations (solid lines) agree well with the recorded data (shaded). The experimental parameters here are ${\mathrm{\Omega }}_p=2\pi \times 5.0\mathrm{MHz}$, ${\mathrm{\Omega }}_c=2\pi \times 8.7\mathrm{MHz}$, ${\mathrm{\Delta }}_p=2\pi \times 90\mathrm{MHz}$, ${\gamma }_{12}=2\pi \times 0.015\mathrm{MHz}$, and $\mathrm{\Delta }k=97.2\mathrm{rad}/\mathrm{m}$.

Figure 4

Evolution of anti-$PT$ supermodes in terms of normalized output powers as functions of atomic density $N$ or $\delta$ at two-photon resonance ($\mathrm{\Delta }\omega =0$) under different seeding. The solid lines are theoretical curves, and circles and squares are the experimental data. The shaded areas are anti-$PT$ phase broken (light orange) and phase-unbroken (light purple) regimes. The dashed lines in (a2) and (b2) are numerical simulations for a medium with a longer length ($10L$) so as to explicitly reveal the abrupt phase transition. Other involved parameters are same as those in Fig. 3.

Figure 5

Re $[{\lambda }_{\pm }]$ and $\mathrm{Im}[{\lambda }_{\pm }]$ of the two eigenpropagation constants of paired Stokes–anti-Stokes supermodes as a function of trapped atomic density $N$ or $\delta$ subject to different seeding operations.