Optical homodyne detection

Interference of a signal with a coherent reference at the same optical carrier frequency to measure phase-sensitive field information. The reference fixes an optical quadrature. The inferred noise depends on reference strength, detector efficiency and the detection arrangement.

A coherent reference field that selects the measured optical quadrature through its relative phase. Mode matching and phase stability determine how faithfully the desired signal optical quadrature is sampled.

Optical homodyne tomography reconstructs an optical quantum state from repeated optical quadrature measurements at different phases. A strong coherent local oscillator interferes with the signal on a beam splitter; balanced homodyne detection subtracts the two detector outputs to estimate the quadrature selected by the reference phase.

Many identically prepared signals give a distribution for that quadrature. Repeating at several phases supplies complementary distributions from which a density matrix or optical Wigner function can be estimated. One quadrature distribution alone generally does not determine the state, just as a single projection of a three-dimensional object does not reveal its full shape. Detector efficiency, mode matching and phase noise must be included in the measurement model; otherwise apparent loss or excess noise can be assigned incorrectly to the signal state.

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.