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 $c=(a+ib)/\sqrt{2}$ and $d=(ia+b)/\sqrt{2}$, the difference is $I=c^{†}c-d^{†}d=i(a^{†}b-a{b}^{†})$. A reference $\beta =|\beta |e^{i\varphi }$ gives mean $2|\beta |⟨X_{\vartheta }⟩$ with $\vartheta =\varphi +\pi /2$. 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.

For the convention of balanced homodyne detection, assume independent signal and coherent reference states and ideal detectors. Write $b=\beta +\delta b$, with $\delta b$ in vacuum. The current difference splits into $2|\beta |X_{\vartheta }$ and $i(a^{†}\delta b-a\delta b^{†})$. Cross terms have zero expectation, and the second term has variance $⟨a^{†}a⟩$. Consequently the exact variance is $4|\beta {|}^2\mathrm{Var}{X}_{\vartheta }+⟨a^{†}a⟩$. Division by $4|\beta {|}^2$ approaches the signal optical quadrature variance when the photon-number correction is negligible.