Cooperativity

The square of the coupling constant is $g^2= \frac{\text{d}^2 \omega }{2 V_{\text{mode}} \epsilon_{0} \hbar },$ where d is a dipole moment matrix coefficient, $\epsilon_{0}=8.854\cdot10^{12}$ F/m, $\hbar =2\pi\cdot6.626 \cdot10^{34}$. The transition frequency for Rb87 D1 line is $\omega =2\pi\cdot 377.107\cdot 10^{12}$. The mode volume we assume to be $V_{\text{mode}} = 2 w(0)l$, where w is the waist, l is the cavity length. Thus the cooperativity is $C=\frac{4Ng^2}{\kappa \gamma}$, the number of atoms could be found in Optical_depth. The comparison between our measurement on D1 line and Bimbard's measurement on D2 line, closed transition.

Gaussian beam width $w(z)$ as a function of the axial distance $z$ . $w_0$: beam waist; $b$: confocal parameter; $z_\mathrm{R}$: Rayleigh length; $\Theta$: total angular spread

	C	dipole moment, d	$\frac{g}{2 \pi}$ [MHz]	$\frac{g^2}{4\pi^2}$	$\gamma/(2\pi)$ [MHz]	$\kappa/(2\pi)$ [MHz]	$\frac{g^2}{\kappa \gamma}$	Number of atoms	Finesse
our	10		0.0138		5.746	15	2.2*10^-6	1.13*10^6	100
Bimbard	10		0.016		5.746	15	2.98*10^-6	0.8*10^6
Bimbard article	15 ($C=\frac{Ng^2}{\kappa \gamma}$)		0.3*		D1 line	10	0.0015	10^4	120

• found from other values from article

Cooperativity measurement

To overlap atomic cloud with a cavity mode and characterize an interaction strength, we measured the cooperativity parameter mentioned above (the ratio between the strengths of coherent and incoherent processes). We sent two lasers into the cavity. One of them was an MBR laser (violet colour on a picture \ref{fig:coop_measurement}). It passed through an acousto-optic modulator (AOM) in order to achieve two beams with different frequencies. The zeroth order beam served as a local oscillator (LO) for a heterodyne measurement. The first order was sent into the cavity to probe the atomic cloud. Thus it was set to be on resonance with Rb 87 D1 line, "$F=2 \rightarrow F'=1$". When it leaks out from the cavity it reflects into a heterodyne detector. The second laser is homemade ECDL (yellow color on a picture \ref{fig:coop_measurement}) and is used to lock the cavity frequency via PDH technique \citep{drever1983laser} while being phase-locked to MBR by homemade lock \citep{macrae2009phaselock}. There are two requirements on ECDL central frequency. The laser should not effect the atoms and at the same time it should be in resonance with the cavity. Simultaneously cavity itself should be in resonance with the probe laser. Thus we lock ECDL to MBR laser 2 FSRs away (6.8 GHz). After these preliminary steps we acquire data points $\frac{I{refl}}{I_{in}}$. The presence of the atoms will affect reflected intensity. Non-zero imaginary part of susceptibility increases intra-cavity losses, while the real part shifts resonance frequency. We estimated our cooperativity to be $C \approx 10$.

File:Coop measurement.pdf

Signal from the photodetector. Deep is due to absorption in atoms.