Collective State Measurement of Mesoscopic Ensembles with Single-Atom Resolution

We demonstrate single-atom resolution, as well as detection sensitivity more than 20 dB below the quantum projection noise limit, for hyperfine-state-selective measurements on mesoscopic ensembles containing 100 or more atoms. The measurement detects the atom-induced shift of the resonance frequency of an optical cavity containing the ensemble. While spatially varying coupling of atoms to the cavity prevents the direct observation of a quantized signal, the demonstrated measurement resolution provides the readout capability necessary for atomic interferometry substantially below the standard quantum limit and down to the Heisenberg limit.

Figure 1

Experimental setup. (a) Atoms are confined in a cavity by far-off-resonant 852 nm trap light, while a near-detuned probe beam is used to determine the cavity resonance frequency, which has been shifted by the atoms. (b) Pound-Drever-Hall signal produced by the heterodyne detection method (blue curve, left). The atoms’ index of refraction shifts the cavity resonance by an amount proportional to the atom number $N$ (red curve, right). (c) Simplified level structure of ${}^{87}\mathrm{Rb}$, showing the $|F=2⟩\to |F^{\prime }=3⟩$ probe light and the $|F=1⟩\to |F^{\prime }=2⟩$ repumping light for total atom number measurements.

Figure 2

Measured atom number variance as a function of integration time for $N=110$ atoms in $F=2$ (red squares), $N=30$ atoms in $F=2$ (blue triangles), and an empty cavity with no atoms (black diamonds). The probe detuning is $\Delta /(2\pi )=250\mathrm{MHz}$. The solid lines are fits according to Eq. (1). Inset: typical signal trace. At $t=0$, we introduce the probe laser to the cavity and observe a Pound-Drever-Hall signal. We integrate the signal over two periods of equal length $\tau$ and take the difference $N-N^{\prime }$ to find the variance in atom number.

Figure 3

Detection variance as a function of atom number for probe-atom detunings of $\Delta /(2\pi )=250\mathrm{MHz}$ (red circles) and $\Delta /(2\pi )=375\mathrm{MHz}$ (blue squares). Open (closed) symbols correspond to hyperfine-state-sensitive detection (total atom number detection). The shaded region indicates single-atom resolution, $(\Delta N{)}^2<1$. Inset: comparison of the measured detection variance to the projection noise, $(\Delta N{)}^2=N$, that would be observed for uniformly coupled atoms. At $\Delta /(2\pi )=375\mathrm{MHz}$, our detection is 20 dB below the projection noise already for 100 atoms.

Figure 4

Observation of the variance of a binomial distribution and verification of single-atom signal. (a) For an ensemble of $N=250$ atoms with cavity shift $\delta \omega$, a fraction $f$ is removed on average by optical pumping. The observed change in cavity shift, $\delta {\omega }_d$, is measured, and the normalized variance $V=\mathrm{Var}(\delta {\omega }_d)/\delta \omega$ is plotted as a function of $f$. The dashed line is a model prediction with no free parameters of the form $V=\frac{3}{4}\alpha (g_0^2/\Delta )f(1-f)$, with $\alpha =0.94$ (see text), and confirms the cavity shift ${\omega }_0=g_0^2/\Delta$ per atom. (b) The extracted value of the single-photon Rabi frequency $2g_0$ from each point agrees with the value in our system calculated from first principles and independently measured cavity parameters (dashed line).