Quantize one optical mode with explicit normalization

Choose one transverse vacuum normal mode with frequency omega and a vector profile u. Normalize the volume integral of its squared magnitude to one. The profile therefore has units of inverse length to the three-halves power and includes polarization. A uniform traveling-wave mode in a periodic box has unit polarization vector divided by square root of volume, with a spatial phase. A standing-wave cavity uses a different normalized profile satisfying its conducting boundaries.

With this convention the positive-frequency electric operator is proportional to u times the annihilation operator; its Hermitian conjugate supplies the negative-frequency part. An overall mode phase is conventional and can move the displayed factor of i into the operator. A separate factor of two for polarization must not be inserted into one selected mode. Other independent polarizations are additional modes, each with its own oscillator and normalization. Material dispersion or absorbing dielectrics require a more elaborate energy normalization.

The mode Hamiltonian is the harmonic-oscillator Hamiltonian with energy spacing hbar omega. The number operator has nonnegative integer eigenvalues, and the lowering operator removes one excitation with a square-root matrix element. The ground state has zero photons but nonzero zero-point energy. Adding one photon always increases energy by the same spacing in this ideal linear mode. A nonlinear cavity can have different spacings, so the simple oscillator description is a model assumption.

A number eigenstate has exact excitation number and zero mean field amplitude. It can nevertheless have substantial electric-field fluctuations and energy. Mean field, mean photon number, and a particular photon-count outcome are different quantities. A state can also be a superposition of number states, in which case repeated ideal number measurements produce a distribution. The amplitude coefficients and the chosen measurement determine that distribution; one cannot read a definite photon count from an oscillating expectation value alone.

Define two Hermitian quadratures by real and imaginary combinations of the ladder operators divided by square root of two. Their commutator is i, and the product of their standard deviations is at least one half. The vacuum has equal variances of one half and saturates this bound. Another common convention divides by two and gives variance one quarter instead; both are valid, but mixing the formulas produces incorrect squeezing and detector calibrations.

For a number state, each quadrature variance is occupation plus one half, while both quadrature means vanish. The two-state lab can illustrate only the probabilities in a deliberately restricted zero-or-one-photon superposition. It does not implement the infinite oscillator ladder, field normalization, or a quadrature distribution. Its relative-phase playback preserves basis probabilities. Treat that as geometry within a two-dimensional slice, not as a cavity calculation or a photon-counting detector.