Extreme nonlinear response of ultranarrow optical transitions in cavity QED for laser stabilization

We explore the potential of direct spectroscopy of ultranarrow optical transitions of atoms localized in an optical cavity. In contrast to stabilization against a reference cavity, which is the approach currently used for the most highly stabilized lasers, stabilization against an atomic transition does not suffer from Brownian thermal noise. Spectroscopy of ultranarrow optical transitions in a cavity operates in a very highly saturated regime in which nonlinear effects such as bistability play an important role. From the universal behavior of the Jaynes-Cummings model with dissipation, we derive the fundamental limits for laser stabilization using direct spectroscopy of ultranarrow atomic lines. We find that, with current lattice clock experiments, laser linewidths of about 1 mHz can be achieved in principle, and the ultimate limitations of this technique are at the 1 $\mu$Hz level.

Figure 1

Schematic of cavity-enhanced ultranarrow-linewidth absorption spectroscopy for laser stabilization. M, mirror; BS, beam splitter; LO, local oscillator; D, photodiode.

Figure 2

Intracavity intensity as a function of in-coupled intensity for $\mathcal{C}=100$ and $\Delta =0$ (blue solid line), $\Delta =10T_2^{-1}$ (purple dashed line), and $\Delta =100T_2^{-1}$ (yellow dotted line). The vertical dashed lines mark the lower and upper thresholds for bistability. The diagonal dashed lines show the intracavity intensity for a completely bleached atomic ensemble, ${\left|a\right|}^2=I_{\mathrm{in}}$, and for the unsaturated limit, ${\left|a\right|}^2=I_{\mathrm{in}}/{\left(2\mathcal{C}\right)}^2$. The saturation photon number is $n_0=\gamma T_2^{-1}/\left(4g^2\right)$.

Figure 3

Intracavity intensity as a function of detuning of the driving laser from the atomic resonance and from the cavity for $\mathcal{C}=100$ and $I_{\mathrm{in}}=5\times {10}^3$. Only the solution with the largest intracavity intensity is shown. Near resonance there are two additional solutions (see Fig. 4). The white hyperbolas indicate the resonances of the weakly driven system.

Figure 4

Intracavity intensity for $\mathcal{C}=100$ as a function of detuning for in-coupled intensities $I_{\mathrm{in}}=5\times {10}^3$ (blue solid line), $1\times {10}^3$ (purple dashed line), and $1\times {10}^2$ (yellow dotted line).

Figure 5

Phase shift of the transmitted cavity light (with respect to the LO) due to the atomic medium inside the cavity and as a function of laser detuning. Here, $\mathcal{C}=100$ and $\beta =2$. The dotted line is the linear approximation for the phase near zero detuning. Inset: Cavity transmission curve for the same parameters.