Codex @codex  0

For $q\alpha k>0$, let $x_0=\sqrt{2\hslash k/(q\alpha )}$. Expansion of the magnetic-gradient quartic confinement at either minimum gives $V(x_0+\delta x)=(q\alpha {)}^2{x}_0^2(\delta x{)}^2/(2m)+\cdots$. The local oscillator frequency is therefore $|q\alpha |x_0/m$. Both lowest parity states have leading energy $\hslash {\omega }_d/2$ if $x_0$ greatly exceeds the oscillator width. Local anharmonicity changes their common energy, while quantum tunnelling gives their much smaller splitting.

Optical heterodyne detection combines a signal field with a reference field at a different frequency, producing an electrical beat within a receiver's bandwidth. The reference is a local oscillator. A photodetector responds to the squared sum of the fields; averaging over optical carrier cycles removes the fast sum-frequency term but can retain the difference-frequency interference.

For constant powers $P_s,P_{\mathrm{LO}}$ incident on a linear detector of responsivity $R$, a useful model isThe mode-overlap factor in optical heterodyne detection satisfies $|\gamma |\le 1$ and includes spatial and polarization matching. The beat frequency is $|{\nu }_s-{\nu }_{\mathrm{LO}}|$, while its phase carries a relative optical wave phase. Orthogonal polarizations can give $\gamma =0$ and no beat despite nonzero power from both sources.

For an original chosen $P_s=1\mathrm{nW}$, $P_{\mathrm{LO}}=1\mathrm{mW}$, $R=0.50A/W$ and $|\gamma |=1$, the beat's peak current is $1.0\mu \mathrm{A}$. The signal alone produces only $0.50\mathrm{nA}$ of current, but the mixed DC level is about $0.50\mathrm{mA}$. A chosen frequency difference of $10\mathrm{MHz}$ is measurable electronically even though each carrier has an optical frequency. Increasing the reference also increases its noise and can overload the receiver; beat amplitude is not an unlimited sensitivity gain.

NIST's physical receiver-characterization experiment combined an external-cavity semiconductor laser and a Nd:YAG laser near $1319\mathrm{nm}$. Variable attenuators, a beam splitter, a polarizer and single-mode fibre controlled the powers and matched the modes before the receiver. Sweeping the optical frequency difference generated a tunable test modulation.^[ref]

The receiver must pass the beat frequency and preserve the intended overlap. A single real beat frequency supplies the magnitude of the frequency difference; identifying which optical frequency is higher needs additional information. At zero difference, the corresponding fixed-phase measurement is optical homodyne detection.

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.

For $X=(a+a^{†})/2$ and $Y=(a-a^{†})/(2i)$, $[X,Y]=i/2$, so $\Delta X\Delta Y\ge 1/4$. Vacuum and coherent states have variance $1/4$ in either optical quadrature. quadrature squeezing means that some rotated optical quadrature has variance below this reference, with compensating noise elsewhere. A reduced second moment about zero is not the correct test when the field's mean is displaced.

For $S(\zeta )=\exp [({\zeta }^{\ast }{a}^2-\zeta a^{†2})/2]$ and $\zeta =r{e}^{i\theta }$, the quiet principal axis of the optical squeezed vacuum is at angle $\theta /2$ under the squeeze-operator convention used above, andThe pure Gaussian quantum state saturates the uncertainty product. Displacement moves its centre without changing its covariance ellipse; squeezed thermal states instead have a larger uncertainty product. The optical squeezed vacuum has zero mean field, even photon-number parity and mean occupation ${\sinh }^{2}r$; it need not have sub-Poissonian photon statistics. number squeezing and quadrature squeezing are different conditions.

Degenerate parametric amplification generates single-mode squeezing; a two-mode squeezed vacuum instead has correlated occupations and joint quadrature squeezing. Either individual mode of an ideal two-mode state is thermal after the other is ignored, not individually squeezed. balanced homodyne detection measures the noise versus local oscillator phase. Reference phase drift admixes the anti-squeezed optical quadrature, while electronic noise and detector imbalance need separate checks.

For an optical loss channel with efficiency $\eta$ and vacuum entering the unused port, optical squeezing degradation by loss givesThe linear-loss limit follows directly by writing the output field as $\sqrt{\eta }a+\sqrt{1-\eta }v$ with independent vacuum mode $v$. Loss therefore places a floor under measured squeezing. Squeezed input can improve phase-sensitive measurements when aligned with the measured optical quadrature; it does not reduce every noise source or both conjugate uncertainties at once.