 Lognormal bias of a heterodyne overlap estimate

Lognormal bias of a heterodyne overlap estimate is the upward mean shift that remains when a zero-mean Gaussian logarithmic error is exponentiated. It does not undo common-gain cancellation in heterodyne overlap uncertainty: a perfectly shared gain cancels in every reading, while residual mismatched gains can still fluctuate.

Write the normalized positive estimate as $\hat{\gamma }=\gamma e^d$, with fixed true overlap magnitude $\gamma$ and $d\sim N({\mu }_d,v_d)$. The moment-generating function of the normal distribution givesThese are exact lognormal distribution identities under the stated error model. When ${\mu }_d=0$, the logarithmic estimate is centered correctly and the median is $\gamma$, but the arithmetic mean exceeds it by relative amount $e^{v_d/2}-1$.

For an original deliberately visible example, $\gamma =0.8$ and logarithmic standard deviation 0.2 give $v_d=0.04$ and mean about 0.81616: a 2.02% mean shift. The values are chosen statistical parameters, not uncertainties reported for a receiver. An estimate can also exceed the physical consistency bound for separately measured heterodyne arm powers; the noisy readout is not itself a physical overlap above one.

If ${\mu }_d$ and $v_d$ are known, multiplying the estimate by $e^{-{\mu }_d-v_d/2}$ corrects its mean in this model. Uncertain variance in heterodyne lognormal bias correction matters when those parameters are estimated: plugging in a noisy variance, especially from related data, is not automatically an exact correction.

Averaging logarithms has a different interpretation from averaging overlaps. For $n$ independent zero-mean log-errors of variance $v_d$, the geometric mean is $\gamma e^{\overline{d}}$, whose arithmetic expectation is $\gamma e^{v_d/(2n)}$. A gain component shared across the entire run does not acquire the independent-shot $1/n$ variance reduction.

The physical calibration paths determine the logarithmic covariance. NIST's heterodyne apparatus controlled the optical modes while separately correcting electrical loss and impedance mismatch.^[ref] A first-order uncertainty calculation alone is not this exact distributional model; nonlinear effects and correlated inputs must be assessed explicitly.^[ref] State the assumed distribution and the intended summary rather than treating median, geometric mean and arithmetic mean as interchangeable.