Preparation of Ultracold Atom Clouds at the Shot Noise Level

We prepare number stabilized ultracold atom clouds through the real-time analysis of nondestructive images and the application of feedback. In our experiments, the atom number $N\sim 10^6$ is determined by high precision Faraday imaging with uncertainty $\mathrm{\Delta }N$ below the shot noise level, i.e., $\mathrm{\Delta }N<\sqrt{N}$. Based on this measurement, feedback is applied to reduce the atom number to a user-defined target, whereupon a second imaging series probes the number stabilized cloud. By this method, we show that the atom number in ultracold clouds can be prepared below the shot noise level.

Figure 1

(a) Experimental sequence. An evaporatively cooled cloud is probed by two series of Faraday images, where rf loss pulses between $F1$ and $F2$ remove a controllable fraction of the atom cloud. (b) Experiment schematic. Faraday imaging probes a cold atom cloud held in a magnetic trap. The images are processed in real time, and the outcome can be used to determine the fraction of atoms removed by rf loss, thereby producing a number stabilized cloud.

Figure 2

(a) Signal sum ${\mathrm{\Sigma }}_S$ vs image number for $F1$. Black lines show traces for 10 representative runs with imaging pulse duration $t=0.66\mathrm{ms}$, and the dashed red line represents the mean trace $⟨{\mathrm{\Sigma }}_S⟩$ for the entire data set. (b) Error $E$ for the same runs as (a). (c) Relative uncertainty vs $t$. Red circles represent $⟨{\sigma }_{\mathrm{\Sigma }}⟩$, the solid red line represents the fit of the imaging model ${\sigma }_{\mathrm{Mod}}$, and the dash-dotted red line represents the mean atom loss contribution to ${\sigma }_{\mathrm{Mod}}$ fit $\propto t$. Black squares represent $⟨{\sigma }_E⟩$, the solid black line represents the fit of ${\sigma }_{\mathrm{Mod}}$, the dotted black line represents the light shot noise contribution to ${\sigma }_{\mathrm{Mod}}$ fit $\propto 1/\sqrt{t}$, and the dashed black line represents the constant technical noise contribution to ${\sigma }_{\mathrm{Mod}}$ fit. Error bars denote the standard error of the mean over several experimental runs.

Figure 3

The black squares represent the signal sum relative uncertainty vs atom number at $F2$, the solid black line represents the total noise model, the blue diamonds represent the imaging noise contribution, the dashed blue line represents the fit $\propto 1/N$ to imaging noise data, and the dash-dotted black line represents the stochastic atom loss noise contribution $\sqrt{(1-p)/⟨N⟩}$. Error bars were obtained by bootstrapping the data set. Inset: Correlation of error at $F1$ and $F2$. The black squares represent the correlation data with no applied loss, the blue line represents the quadratic fit to correlation data, and the red circles represent the feedback stabilized points.

Figure 4

The red circles represent the signal sum relative uncertainty vs atom number at $F2$ for feedback stabilized clouds. Other lines same as Fig. 3. The shaded region represents the interval where the relative uncertainty in the signal sum lies below the shot noise level, i.e., $1/\sqrt{N}$ expressed as signal sum [27].