Collision-resolved pressure sensing

While a continuous Brownian description of noise from heat and pressure is adequate to model measurements with relatively long integration times, these forces are ultimately generated by quantized degrees of freedom like phonons and gas particles. Fundamentally, the ultimate limit of this sensing problem is to resolve all of the individual system-sensor collisions. Here we propose the use of nanomechanical devices operated with impulse readout sensitivity around the standard quantum limit to sense ultralow gas pressures by directly counting the individual collisions of gas particles on a sensor. We illustrate this in two paradigmatic model systems: an optically levitated nanobead and a tethered membrane system in a phononic band-gap shield.

Figure 1

Schematics of the basic detection scheme, with either a levitated nanoparticle or tethered membrane in the unit cell of a phononic band-gap shield [34, 35]. When an environmental gas particle collides with the mechanical element, it deposits momentum $\mathrm{\Delta }p$, which can be detected by continuously monitoring the position $x\left(t\right)$ of the element.

Figure 2

Example spectrum of collision events, expressed as a differential rate $d\mathrm{\Gamma }$ per given impulse value $\mathrm{\Delta }p$. The black lines label the nominal detection threshold $\mathrm{\Delta }{p}_{\min }=\mathrm{\Delta }{p}_{\mathrm{SQL}}$, with a solid sphere of radius $50\mathrm{nm}$, trapped at either ${\omega }_s/2\pi =1\mathrm{kHz}$ (left) or $100\mathrm{kHz}$ (right). We again assume the gas is dominated by diatomic hydrogen at $300\mathrm{K}$, and we show the predictions for pure hard-sphere scattering (specular) as well as diffusive scattering corrections.

Figure 3

Diffuse momentum cutoff ${\eta }_d$ for a tethered sensor as a function of $T_s$ for various values of ${\eta }_s$.

Figure 4

Diffuse momentum cutoff ${\eta }_d^{\prime }$ for a nanosphere as a function of $T_s$ for various values of ${\eta }_s^{\prime }$.

Figure 5

Fractional pressure uncertainty ${(\delta P/P)}_{\alpha }$ due to uncertainty in $\alpha$ as a function of $\alpha$ for a tethered sensor at various values of ${\eta }_s$.

Figure 6

Fractional pressure uncertainty ${(\delta P/P)}_{\alpha }$ due to uncertainty in $\alpha$ as a function of $\alpha$ for a nanosphere at various values of ${\eta }_s^{\prime }$.