Exceptional Point Superradiant Lasing with Ultranarrow Linewidth

Abstract

Achieving superradiant lasing with an ultranarrow linewidth is crucial for enhancing atomic clock stability in quantum precision measurement. By employing the exceptional point (EP) property of the system, we demonstrate theoretically superradiant lasing with linewidths in the $μ$Hz range, sustained at the high-power level. This is achieved by incoherently pumping optical lattice clock transitions with ultracold alkaline-earth strontium-87 atoms in the EP of a $\mathcal{PT}$-symmetric system. Physically, the atomic coherence reaches a maximum in the EP, significantly amplifying the superradiance effect and resulting in superradiant lasing with an ultranarrow linewidth. This linewidth is even three orders of magnitude smaller than that of superradiant lasing in the systems without EP. Our work extends the realm of superradiant lasing by introducing the EP property, and offers promising applications for developing atomic clocks with exceptional stability and accuracy.

Figures (9)

• Figure 1: (a) Thousands of $^{87}{\rm Sr}$ atoms are trapped in an optical cavity coupled to an auxiliary empty cavity with coupling strength $G$. (b) The level structure of $^{87}{\rm Sr}$: the dipole-allowed $^1S_0$ to $^1P_1$ transition is used for the initial laser cooling. After excitation, not all atoms return to the ground state. A fraction of the atoms populate the $^1D_2$ state, which subsequently decays to the metastable $^3P_2$ and $^3P_1$ states. Atoms in the $^3P_1$ state can spontaneously decay to $^1S_0$ state because the spin-orbit coupling and hyperfine interactions in $^{87}{\rm Sr}$ enable weakly dipole-forbidden (doubly forbidden) transitions, such as $^1S_0$ - $^3P_1$($^1S_0$ - $^3P_0$). Atoms in $^3P_2$ state, which are unable to decay to $^1S_0$ state due to selection rules, are repumped to the $^3S_1$ state, where atoms undergo spontaneous emission with fixed branching ratios to the $^3P_2$, $^3P_1$ and $^3P_0$ states. An additional repumping laser is employed to transfer atoms from $^3P_0$ back to $^3S_1$ due to the relatively long lifetime of the $^3P_0$ state. This scheme continuously maintains the population of atoms in the $^3P_1$ state, from which atoms transition back to the ground state, thereby establishing a closed cycle that enhances cooling efficiency. The dipole-forbidden $^1S_0$ to $^3P_1$ transition is employed for final cooling into the optical lattice. Superradiant lasing is observed on the dipole-forbidden $^1S_0$ to $^3P_0$ transition at 698 $\mathrm{nm}$. (c) The real and imaginary parts of eigenvalues of the tunneling-coupled subsystem versus $G$. The shaded areas and other areas represent the $\mathcal{PT}$-symmetry-breaking phase ($\mathcal{PT}$BP) and $\mathcal{PT}$-symmetry phase ($\mathcal{PT}$SP), respectively. System parameters are $\Delta_a=\Delta_b=0$, $\kappa_a/2\pi=160\,{\rm kHz}$, and $\kappa_b/2\pi=1\,{\rm kHz}$.
• Figure 2: (a) Steady-state excited-state population of an individual atom $\langle \sigma_{1}^{+}\sigma_{1}^{-}\rangle$ and (b) Atom-atom correlation $\langle \sigma_{1}^{+} \sigma_{2}^{-}\rangle$ versus $\eta$ in the $\mathcal{PT}$BP ($G/2\pi=3.975$ kHz, $C_2/2\pi=16.6$ mHz), EP ($G/2\pi=39.75$ kHz, $C_1/2\pi=0.571$ mHz), and (inset) $\mathcal{PT}$SP ($G/2\pi=3975$ kHz). (c) Atom-atom correlation $\langle \sigma_{1}^{+} \sigma_{2}^{-}\rangle$ versus $G$ when $\eta/2\pi=18$ Hz. The shaded region corresponds to the $\mathcal{PT}$BP. (d) Steady-state Dicke state population with varying $\eta$ in the $\mathcal{PT}$BP, EP, and $\mathcal{PT}$SP (left to right). The results are shifted for clarity and the thin black lines mark Dicke ladder boundaries. System parameters are $N=10^{7}$, $\Delta_a=\Delta_b=0$, $g/2\pi=2.41\,{\rm Hz}$, $\kappa_a/2\pi=160\,{\rm kHz}$, $\kappa_b/2\pi=1\,{\rm kHz}$, $\gamma/2\pi=1\,{\rm mHz}$, and $\gamma_{\phi}/2\pi=1\,{\rm mHz}$.
• Figure 3: (a) Steady-state intracavity photon number $\langle a^{\dagger}a\rangle$ and (b) emission linewidth $\Delta\nu$ versus $\eta$ in the $\mathcal{PT}$BP, EP, and $\mathcal{PT}$SP. The solid lines in (b) correspond to the analytical approximate solutions from Eq. (\ref{['eq5']}). (c) $\langle a^{\dagger}a\rangle$ and $\Delta\nu$ versus $G$ for $\eta/2\pi=18$ Hz. The shaded region corresponds to the $\mathcal{PT}$BP. Other system parameters are the same as in Fig. \ref{['fig2']}.
• Figure 4: (a,b) Emission linewidth $\Delta\nu$ and (c,d) output power $P$ in the steady-state versus $\eta$ and $N$ for (a,c) EP and (b,d) $\mathcal{PT}$BP. The white and red dashed lines denote the thresholds for the superradiant lasing. Other system parameters are the same as in Fig. \ref{['fig2']}.
• Figure S1: (a) Dicke state population, (b) atom-atom correlation $\langle\sigma_1^+\sigma_2^-\rangle$ in the steady-state versus $\eta$ at the EP. System parameters are $N={10}^7$, $\Delta_a=\Delta_b=0$, $g/2\pi=2.41\,\mathrm{Hz}$, $\kappa_a/2\pi=160\,\mathrm{kHz}$, $\kappa_b/2\pi=1\,\mathrm{kHz}$, $\gamma/2\pi=1\,\mathrm{mHz}$, and $\gamma_{\phi}/2\pi= 1\,\mathrm{mHz}$.