Feasibility of squeezing measurements with cavity-based atom detection

We numerically analyze the quantum efficiency and dark noise of a cavity-based single-atom detector, with particular emphasis on the ability to measure number squeezing in an atom-laser beam. We consider the influence of the electric-dipole force on an atom in a red-detuned detection beam and discuss the much improved detection efficiency for detuned probe beams, with respect to resonant probes, resulting from this influence. Cavities allow real-time monitoring of atomic flux, with single-atom resolution, but they are much slower than their analog in photonics (the avalanche photodiode), so flux limits must be imposed. The proposed detector operates at a maximum flux of 5000 atoms/second, but with a shot-noise clearance of up to 23 dB, allowing the full advantage afforded by number squeezing to be observed.

Figure 1

(a) Schematic diagram (not to scale) of cavity-detection setup. (b) Hypothetical photon counts for atom-detection events. The dashed lines indicate possible threshold levels for signal classification.

Figure 2

Measured and actual variance of atoms for one second of measurement time as a function of atomic flux. The inset indicates the atomic flux at which these values begin to differ significantly. The dashed line indicates deviation of $5\%$ from the actual variance.

Figure 3

Atom trajectories through the resonator mode. The conditions modeled are for (a) resonant and (b) detuned detection, with ${\Delta }_a=25\gamma$. The atom entrance velocity is $\mathbf{v}=-1(m/s)\widehat{\mathbf{y}}$. Blue traces are trajectories of those atoms entering the mode directly above an antinode and on-axis in the radial direction, while red traces show atomic transits for atoms entering the cavity an axial distance of $\lambda /4$ from a node. Ellipses represent the cavity mode at positions where the field intensity is half its maximum. Note the vastly different scales on the plot axes: the cavity waist is $w_0=20\mu$m, while the standing-wave structure is on the order of the wavelength used ($\lambda \lesssim 1\mu$m).

Figure 4

(a), (b) Histograms of transmitted photon number measured at the cavity output during an atom transit (blue solid line) and for an empty cavity (red dashed line). The shaded regions represent a discriminating detection threshold two standard deviations below the empty-cavity mean photon number. (c), (d) Variation in quantum efficiency $\eta$ (solid line, left axis) and false-count fraction or noise (dotted line, right axis) as a function of discriminating threshold position. The quantum efficiency increases as the discriminating threshold is raised, as does the false-count rate. For a given quantum efficiency, the detuned probe yields a lower false-count rate than the resonant probe.

Figure 5

Dark-noise clearance for a detection integration time of $\tau =20\mu$s and an atomic flux of 5000 atoms/s. The cavity waist is $w_0=20\mu$m. (a) shows a variation of discriminating position and (b) shows optical-mode overlap. The dashed line in (b) corresponds to the waist ratio used in (a).

Figure 6

SNR for different squeezing levels and varying positions for the signal discrimination threshold. The squeezing level is parameterized with the normalized variance $V_S$, as defined in Sec. 3a. The “discriminating threshold” represents a number of standard deviations below the empty-cavity photon number. (a) Resonant and (b) detuned detection conditions are the same as Fig. 3.