= Balanced homodyne detection
Mix a signal mode $a$ and an independent coherent reference mode $b$ on a balanced <beam splitter>, detect both outputs, and subtract the currents. For $c=(a+ib)/\sqrt2$ and $d=(ia+b)/\sqrt2$, the difference is $I=c^\dagger c-d^\dagger d=i(a^\dagger b-ab^\dagger)$. A reference $\beta=|\beta|e^{i\phi}$ gives mean $2|\beta|\langle X_\vartheta\rangle$ with $\vartheta=\phi+\pi/2$. Other beam-splitter conventions shift this phase calibration. The exact <finite-reference balanced homodyne variance> quantifies the strong-reference approximation.
Matched subtraction cancels the shared total-power term. It therefore suppresses common reference intensity noise, while imbalance and phase fluctuations reduce this advantage. Repeated samples measure a distribution, not just its mean; scanning reference phase supports <optical homodyne tomography>.
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