Nonlinear spectroscopy of Sr atoms in an optical cavity for laser stabilization

We study the nonlinear interaction of a cold sample of ${}^{88}\mathrm{Sr}$ atoms coupled to a single mode of a low finesse optical cavity in the so-called bad cavity limit, and we investigate the implications for applications to laser stabilization. The atoms are probed on the weak intercombination line $|5s^2{}^1{S}_0\rangle -|5s5p{}^3{P}_1\rangle$ at 689 nm in a strongly saturated regime. Our measured observables include the atomic induced phase shift and absorption of the light field transmitted through the cavity represented by the complex cavity transmission coefficient. We demonstrate high signal-to-noise-ratio measurements of both quadratures—the cavity transmitted phase and absorption—by employing frequency modulation (FM) spectroscopy (noise-immune cavity-enhanced optical-heterodyne molecular spectroscopy). We also show that when FM spectroscopy is employed in connection with a cavity locked to the probe light, observables are substantially modified compared to the free-space situation in which no cavity is present. Furthermore, the nonlinear dynamics of the phase dispersion slope is experimentally investigated, and the optimal conditions for laser stabilization are established. Our experimental results are compared to state-of-the-art cavity QED theoretical calculations.

Figure 1

(a) Experimental setup. A sample of cold ${}^{88}\mathrm{Sr}$ atoms trapped in a standard magneto-optical trap (MOT) is located inside a low finesse optical cavity $\left(F=85\right)$. The cavity is locked to resonance with the light using the Hänsch-Couillaud [25] scheme. We perform cavity-enhanced FM spectroscopy (NICE-OHMS) where the light is modulated at $\mathrm{\Omega }/2\pi =\text{FSR}$ (free spectral range of the cavity). The detected signal is split into two arms where the $\mathrm{\Omega }$ and $2\mathrm{\Omega }$ components are selected by band-pass filters and separately demodulated using rf mixers. One part of the signal is demodulated at $\mathrm{\Omega }$ giving a phase signal, and the other part is demodulated at $2\mathrm{\Omega }$ giving an attenuation signal. Both the phase shift and the absorption line shape of the transmitted probe light are recorded simultaneously with this scheme. (b) Energy levels of the ${}^{88}\mathrm{Sr}$ atom important for this work. The singlet transition $|5s^2{}^1{S}_0\rangle -|5s5p{}^1{P}_1\rangle$ at 461 nm is used to trap and cool the atoms. The narrow intercombination line transition $|5s^2{}^1{S}_0\rangle -|5s5p{}^3{P}_1\rangle$ at 689 nm is used to probe the atoms.

Figure 2

The solid and dashed lines are predictions based on the theoretical model presented in [20, 21]. The dots are data measured by cavity-enhanced FM spectroscopy. All data are averaged values, where approximately five times as many measurements have been performed for data in (b) due to the weaker signal compared to data in (a). The slight asymmetry of the experimental data in both plots is expected to be due to a drift of the relative phase between the light wavefront and the demodulation phase during scan. (a) Frequency scan of the cavity-transmitted phase shift close to resonance for a carrier input power of $240\text{nW}$, a total number of atoms of $N=5.0\times {10}^8$, and an atom temperature of $T=5.0\text{mK}$. The data points are averages of nine measurements. Theoretical values from Eq. (9) are scaled to data. Inset: high-resolution scan of the central region. The vertical axis in this central region corresponds to the total atomic phase shift of the transmitted sideband, and data are scaled to theory. (b) Frequency scans of the absorption line shape of the cavity-transmitted field measured by the NICE-OHMS technique obtained with a demodulation $\mathrm{\Omega }/2\pi =$ 2FSR. The total number of atoms is $N=4.5\times {10}^8$, an atom temperature of $T=5.0\text{mK}$, and a total input power of $310\text{nW}$ (blue line) and $155\text{nW}$ (green line). Theoretical values from Eq. (10) are scaled to data. Inset: high-resolution scans around the central region. The units on the inset axes are the same as in (b).

Figure 3

The input power dependence of the slope of the phase dispersion around resonance and the corresponding shot-noise limited linewidth for a total number of atoms of $N=4.0\times {10}^8$ and MOT temperature of $T=5.0\text{mK}$. The solid lines are theoretical predictions and the dots are experimental data. (a) The slope of the phase dispersion around resonance measured for different input powers. The data and theory are in excellent accordance and the nonlinear dependence is evident. Notice that the power axis is logarithmic. (b) The shot-noise limited linewidth from Eq. (12) for different input powers corresponding to the measured parameters in (a). The solid lines are theoretical predictions and the dots are experimental values. The blue theory line corresponds to a ratio of carrier and sideband input powers of $\frac{P_{\mathrm{sig}}}{2P_{\mathrm{sideband}}}=1.65$ and the green theory line corresponds to $\frac{P_{\mathrm{sig}}}{2P_{\mathrm{sideband}}}=0.75$. (c)–(e) Frequency scans of the phase dispersion for different input powers. All experimental values in (a) and (b) are evaluated by fitting theoretical curves to frequency scans of the phase dispersion around resonance. The nonlinear dynamics of the overall dispersion shape is in very good accordance with the theory.

Figure 4

Theoretical values of the transmission coefficient of the carrier (dashed blue), ${\chi }_0$, the combined line shape of the atomic absorption and the attenuation due to atomic dispersion (green), ${\chi }_0\text{Re}\left({\chi }_2\right)$, and the total cavity transmitted phase shift (red). The black vertical dashed lines highlight the fact that transmission minima occur at dispersion maxima.