Laser stabilization using saturated absorption in a cavity-QED system

We consider the phase stability of a local oscillator (or laser) locked to a cavity-QED system composed of atoms with an ultranarrow optical transition. The atoms are cooled to milli-Kelvin temperatures and then released into the optical cavity. Although the atomic motion introduces Doppler broadening, the standing-wave nature of the cavity causes saturated absorption features to appear, which are much narrower than the Doppler width. These features can be used to achieve an extremely high degree of phase stabilization, competitive with the current state of the art. Furthermore, the inhomogeneity introduced by finite atomic velocities can cause optical bistability to disappear, resulting in no regions of dynamic instability and thus enabling a new regime accessible to experiments where optimum stabilization may be achieved.

Figure 1

Schematic of the cavity-QED experiment with a thermal sample of atoms with Doppler width ${\mathrm{\Gamma }}_d$, each of which have a narrow optical transition of width ${\gamma }_a\ll {\mathrm{\Gamma }}_d$. The atoms are probed with a carrier with frequency ${\omega }_c$, and two sidebands located at ${\omega }_c\pm \mathrm{FSR}$, where FSR is the free spectral range of the cavity. The carrier frequency ${\omega }_c$, which is close to the atomic frequency ${\omega }_a$, is locked on the cavity mode frequency $\omega$, while the sidebands are assumed far off resonance, typically $\sim {10}^4$ atomic linewidths. By demodulating the light transmitted through the cavity at the FSR, detection of the nonlinear phase response of the transmitted light is achieved. This phase response results in a photocurrent that serves as a frequency discriminating error signal that stabilizes the laser frequency through the requirement $\mathrm{\Delta }i=0$.

Figure 2

Development of the extra absorption and dispersion features in the transmission (T) and phase shift $\left(\varphi \right)$, as the input intensity is increased. Blue (dark gray) solid curves are for a standing wave cavity; red (light gray) dashed curves are for a traveling wave cavity. For all plots, $N{C}_0=600$ and ${\delta }_0/{\gamma }_p=260$, which in the case of ${}^{88}\mathrm{Sr}$ corresponds to a temperature of $\sim 15$ mK. (a) ${\left|y\right|}^2=5\times {10}^{-1}$. (b) ${\left|y\right|}^2=6\times {10}^1$. (c) ${\left|y\right|}^2=9\times {10}^2$. The inset is zoomed in to emphasize the central features. (d) Input vs intracavity intensity at resonance. The black dots label the input and intracavity intensities of (a)–(c), and the dashed line is ${\left|y\right|}^2={\left|x\right|}^2$ for reference. Note that there is no bistability.

Figure 3

Linewidth at the optimum ${\left|y\right|}^2$, which gives the smallest linewidth, as a function of included orders of $l$, normalized by the $l=0$ linewidth. Blue circles: $N{C}_0=600,{\delta }_0/{\gamma }_p=260,{\left|y\right|}^2=1270$. Red squares: $N{C}_0=6000,{\delta }_0/{\gamma }_p=260,{\left|y\right|}^2=7700$. Black diamonds: $N{C}_0=600,{\delta }_0/{\gamma }_p=80,{\left|y\right|}^2=850$.

Figure 4

(a) Scaled input $\left({\left|x\right|}^2\right)$ vs intracavity $\left({\left|y\right|}^2\right)$ intensities for several temperatures with $N{C}_0=800$. (b) Slope at resonance for several temperatures with $N{C}_0=800$. (c) Stabilization linewidth as a function of $N{C}_0$ for several temperatures. For ${\delta }_0/{\gamma }_p=0$, the black dashed line is calculated at the fixed input intensity ${\left|y\right|}^2=4N{C}_0$; the black solid line is calculated at the fixed input intensity ${\left|y\right|}^2={\left(N{C}_0\right)}^2/4$. For ${\delta }_0/{\gamma }_p\ne 0$, the linewidth is calculated with the input intensity fixed to the optimum value that gives the smallest linewidth. In the case of ${}^{88}\mathrm{Sr},{\delta }_0/{\gamma }_p=150$ corresponds to a temperature of $\sim 5$ mK.