2D companion to the 3D optomechanical force sensor: the same physics, read off a schematic cavity side-view instead of a rendered scene. A nanobeam of effective mass m coated as one mirror of an optical cavity behaves as a driven-damped oscillator with resonance Ωm and damping γm = Ωm/Q. Its mechanical susceptibility is
χ(Ω) = 1 / [ m(Ω_m² − Ω² − iΩγ_m) ]
Three independent noise sources add in the measured displacement power spectral density Sxx(Ω):
S_xx(Ω) = S_th(Ω) + S_imp/P + S_ba·P·|χ(Ω)|²
S_th(Ω) = 4 k_B T m γ_m |χ(Ω)|² (thermal Langevin force)
S_imp/P (photon shot-noise readout, ∝ 1/power)
S_ba·P·|χ(Ω)|² (radiation-pressure back-action, ∝ power)
Raising the readout power P sharpens the position readout (lower Simp) but pumps more radiation-pressure shot noise into the beam's own motion (higher Sba) — the two curves cross at the power that minimizes total added noise, which by construction bottoms out at the standard quantum limit
S_xx^SQL(Ω_m) ≈ ħ / (m Ω_m)
An equivalent force sensitivity follows from √SFF = √Sxx / |χ(Ωm)|, and the minimum resolvable force over a detection bandwidth Δf is √(SFF·Δf).
- Readout power — slides the imprecision/back-action trade-off; the "vs. SQL" readout shows how close the sensor sits to the quantum limit.
- Bath temperature — scales the thermal-noise floor 4kBTmγm; cryogenic cooling (4 K) suppresses it by ~75× versus room temperature.
- Test force — a coherent sinusoidal force at Ωm that raises a driven peak above the noise floor, exactly how a real signal is distinguished from the noise background.
- Detection bandwidth — the spectrum is integrated over Δf to give the RMS displacement/force actually measured in a finite averaging time.
- Mechanical Q and mass — a higher Q narrows and raises the thermal-noise Lorentzian at resonance; a lighter beam sits closer to the SQL for the same temperature (SxxSQL ∝ 1/m).
Real-world relevance: this trade-off — thermal noise, imprecision, back-action and the SQL — is exactly what governs LIGO's mirrors, atomic-force-microscope cantilevers, and nano-optomechanical single-molecule force sensors.