Manuscript En01-C@1 · En01-C.tex · section · 00a2161c

Exact moment transfer during sensing

Finite squared-position sensing and complete records › Exact moment transfer during sensing · lines 1359–1398

En01-C@1#eq:sensorMoment, En01-C@1#eq:r, En01-C@1#eq:recordlinear

With constant sensor curvature $κ_{s}$ and strength $μ$ , let $m = (v_{x}, c, v_{p})^{⊤}$ . For either quadratic localization or full regularized kicks,

m ˙ = B_{s} m + j_{h} (t), B_{s} = 0 - κ_{s} 8 μ ℏ^{2} 2/ M 0 - 2 κ_{s} 0 1/ M 0, j_{h} (t) = 00 2 ℏ^{2} D_{s, h} (t) .

The measurement term contributes

- μ [x^{2}, [x^{2}, p^{2}]] = 8 μ ℏ^{2} x^{2} .

It contributes zero to the other two moment equations.

Define the sensor row

r^{⊤} = \frac{η _{τ}}{τ} \int_{0}^{τ} e_{x}^{⊤} e^{B_{s} u} d u, e_{x} = (1, 0, 0)^{⊤} .

Then

E_{h} Z = r_{x} v_{x, in} + r_{c} c_{in} + r_{p} v_{p, in} + b_{s, h},

where $b_{s, h}$ is the integrated source term. It can depend on the hypothesis and support-noise amplitude, but is common across settings with identical sensing specifications.

Displayed formulas in this passage

En01-C@1#eq:sensorMoment · line 1364
$m ˙ = B_{s} m + j_{h} (t), B_{s} = 0 - κ_{s} 8 μ ℏ^{2} 2/ M 0 - 2 κ_{s} 0 1/ M 0, j_{h} (t) = 00 2 ℏ^{2} D_{s, h} (t) .$
En01-C@1#eq:r · line 1384
$r^{⊤} = \frac{η _{τ}}{τ} \int_{0}^{τ} e_{x}^{⊤} e^{B_{s} u} d u, e_{x} = (1, 0, 0)^{⊤} .$
En01-C@1#eq:recordlinear · line 1391
$E_{h} Z = r_{x} v_{x, in} + r_{c} c_{in} + r_{p} v_{p, in} + b_{s, h},$
unlabeled · line 1378
$- μ [x^{2}, [x^{2}, p^{2}]] = 8 μ ℏ^{2} x^{2} .$

Exact moment transfer during sensing

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