Quantum sensing#
Quantum limits#
A phase read from \(N\) independent probes repeated \(\nu\) times is limited by the standard quantum limit
\(\Delta\varphi_{\mathrm{SQL}} = 1/\sqrt{N\nu}\); entangled probes can in principle reach the Heisenberg limit
\(\Delta\varphi_{\mathrm{H}} = 1/(N\sqrt{\nu})\) (Giovannetti, Lloyd and Maccone, 2004). phase_sensitivity returns either.
Atom interferometer#
A light-pulse atom interferometer with pulse separation \(T\) and effective wave number \(k_{\mathrm{eff}} = 4\pi/\lambda\) reads a constant acceleration \(a\) as the phase \(\varphi = k_{\mathrm{eff}}\,a\,T^2\) (Kasevich and Chu, 1992). With fringe contrast \(C\) the single-shot acceleration sensitivity is
so it improves as \(1/\sqrt N\) and as \(1/T^2\) (atom_interferometer_sensitivity).
Accelerometer model#
A sensor \(k\) outputs, at its rate \(f_k\), \(z = a + b_k + n\) with white noise \(n \sim \mathcal{N}(0, \sigma_k^2)\) and a random-walk bias
\(b_k(t + \mathrm{d}t) = b_k(t) + w\), \(w \sim \mathcal{N}(0, \beta_k^2\,\mathrm{d}t)\), initial bias \(\mathcal{N}(0, b_{0,k}^2)\) (Sensor).
Dead-reckoning filter#
The true acceleration is an Ornstein-Uhlenbeck process, \(a(t + \mathrm{d}t) = e^{-\mathrm{d}t/\tau}a(t) + \sqrt{1 - e^{-2\mathrm{d}t/\tau}}\,\sigma_a\,\xi\). The Kalman state is \((p, v, a, b_1, \dots, b_K)\) with
process noise \(\sigma_a^2(1 - e^{-2\mathrm{d}t/\tau})\) on \(a\) and \(\beta_k^2\,\mathrm{d}t\) on \(b_k\). Sensor \(k\) updates the filter with \(H = e_a + e_{b_k}\) and \(R = \sigma_k^2\) at its own sample times; a position fix updates with \(H = e_p\) and its own variance (Kalman, 1960). Position error grows when a bias cannot be separated from the true acceleration; a sensor with a much smaller bias makes the other sensor’s bias observable, which is why a slow, stable sensor helps a fast, drifting one.