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Let $b_k=\cosh {\theta }_k{a}_k+\sinh {\theta }_k{a}_{-k}^{†}$ for real even ${\theta }_k$, and let every $b_k$ annihilate a normalized vacuum state. Inverting the Bogoliubov transformation gives $a_k=\cosh {\theta }_k{b}_k-\sinh {\theta }_k{b}_{-k}^{†}$. Only $⟨b_k{b}_q^{†}⟩={\delta }_{kq}$ survives, proving the displayed vacuum expectation values. Taking the Hermitian adjoint gives the same real negative pair amplitude for two creation operators. The normal occupation number is quadratic at small ${\theta }_k$, whereas the pair amplitude is linear. For one opposite-momentum pair the two-mode squeezed vacuum is $\mathrm{sech}\theta {\sum }_{n\ge 0}(-\tanh \theta {)}^n|n,n⟩$, fixing the sign convention directly.

For a weakly depleted homogeneous Bose gas with repulsive contact interaction, set $g=uN/V$ and ${ϵ}_k={\hslash }^2{k}^2/(2m)$. A zero-mode coherent state times a two-mode squeezed vacuum for each opposite-momentum pair has mean particle number $N=N_0+{\sum }_{k\ne 0}{\sinh }^2{\theta }_k$. Substitution of $N_0=N-\sum {\sinh }^2{\theta }_k$ into the leading condensate interaction energy, followed by a consistent quadratic truncation, gives the displayed variational energy. Omitting this substitution loses the subtraction embodied in the number-conserving Bogoliubov quadratic Hamiltonian.

For each mode set $A_k={ϵ}_k+g$ and $E_k=\sqrt{A_k^2-g^2}$. If $A_k>|g|$, choose $\tanh (2{\theta }_k^{\ast })=g/A_k$. The mode contribution equals $\frac{1}{2}{E}_k\cosh [2({\theta }_k-{\theta }_k^{\ast })]-A_k/2$, proving a unique minimum $(E_k-A_k)/2$. At fixed ultraviolet cutoff this lowers the energy relative to the uncorrelated condensate. The approximation requires small quantum depletion of a Bose condensate; an unregulated three-dimensional ground-state energy also requires renormalization of a three-dimensional dilute-gas contact coupling.

Put ${ϵ}_k={\hslash }^2{k}^2/(2m)>0$ for $\mathbf{k}\ne 0$ and $g=u\rho$, using $\rho =N/L^3$. Since the zero mode has zero kinetic energy, the exact kinetic energy in this trial state isTogether with part (d), the fixed-mean-number Bogoliubov variational energy isFor repulsive contact interaction, $u>0$, each ${ϵ}_k+g>g>0$. Using the double-angle hyperbolic-function identities,soThe stationary point therefore satisfiesThis is a minimum, not just a vanishing derivative. Define the positive Bogoliubov quasiparticle dispersionAt the displayed solution ${\theta }_k^{\ast }$, $\cosh 2{\theta }_k^{\ast }=({ϵ}_k+g)/E_k$ and $\sinh 2{\theta }_k^{\ast }=g/E_k$. HenceThe hyperbolic cosine has its unique minimum at zero, proving global minimization of each retained term. Treating equal-magnitude wavevectors together simply multiplies the same function by their multiplicity. At a fixed ultraviolet cutoff, the resulting energy change from $\theta =0$ isIndeed $E_k<{ϵ}_k+g$. More directly, for small positive $\theta$,The pair vacuum expectation value lowers the interaction energy at first order in $\theta$, whereas the nonzero occupation number and its kinetic energy cost start at second order. The two-mode squeezed vacuum admixes opposite-momentum pairs into the condensate, introducing spatial correlations that reduce the chance of two bosons coinciding. These correlations lower repulsive contact interaction enough to outweigh the added kinetic energy. A small quantum depletion of a Bose condensate can therefore lower the energy below the uncorrelated condensate value.

Two qualifications delimit this result. First, the paper does not state the sign of $u$. More generally a finite stable minimum exists only when ${ϵ}_k+g>|g|$ for every retained nonzero mode; repulsion ensures this. For example, with $g=-{ϵ}_k$ in some mode, $f_k(\theta )={ϵ}_k\sinh (2\theta )/2$ tends to $-\infty$ as $\theta \to -\infty$, so an unconditional minimum claim for arbitrary $u$ is false. Also the formula minimizes the truncated Bogoliubov theory energy, not the exact quartic Hamiltonian including its discarded terms, and it must remain in the regime of small quantum depletion of a Bose condensate.

Second, a three-dimensional unregulated contact interaction cannot give a finite absolute beyond-leading-order ground-state energy by this bare sum. For large $k$, $E_k-{ϵ}_k-g=-g^2/(2{ϵ}_k)+O({ϵ}_k^{-2})$, whose three-dimensional sum has an ultraviolet divergence. A finite ultraviolet cutoff, or renormalization of a three-dimensional dilute-gas contact coupling matched to a physical scattering length, is needed. This does not alter the modewise minimization or the finite-cutoff lowering mechanism just derived.

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.

spontaneous parametric down-conversion uses a second-order nonlinear polarization to convert pump energy into correlated signal and idler photons. With a classical undepleted pump, a single-mode interaction-picture model isThe creation term acts on the signal/idler vacuum, so no classical signal seed is required. The pump supplies the emitted energy; vacuum fluctuations are not an extra energy source. In the small-gain limit the SPDC pair state iswith normalization corrections understood at second order. More generally it is a two-mode squeezed vacuum with correlated pair numbers. A classical calculation with both generated fields exactly zero would miss the spontaneous start.

Efficient generation requires ${\omega }_p={\omega }_s+{\omega }_i$ and approximate wave-vector matching ${\mathbf{k}}_p={\mathbf{k}}_s+{\mathbf{k}}_i$. birefringence or a reciprocal grating vector in quasi-phase matching can compensate dispersion. In a continuum-mode treatment the pair amplitude includesThe pump spectrum constrains the sum frequency, while phase matching constrains the allowed frequency combinations and emission directions. type-I phase matching produces daughters in the same polarization class; type-II phase matching produces orthogonal ordinary/extraordinary classes. Correlation of pair numbers or orthogonal polarizations does not alone guarantee a polarization-entangled state in two chosen output paths.

One application is a heralded single-photon source: detect one partner and condition measurements on the other. Low gain suppresses multiple-pair contamination, whose probability begins at fourth order in the gain. Efficient collection and low detector noise matter. Spectral entanglement with an unresolved herald can leave the partner mixed; engineering or filtering the joint spectrum improves purity at a possible cost in brightness. Heralding does not make a photon appear in every pump pulse.

A second application is entangled-pair generation for quantum communication and tests of Bell inequalities. If the assignments $H_s{V}_i$ and $V_s{H}_i$ are indistinguishable except for polarization, their amplitudes can form $(|H_s{V}_i⟩+e^{i\phi }|V_s{H}_i⟩)/\sqrt{2}$. Walk-off, group delay or frequency labels can reveal which assignment occurred and destroy polarization interference. Their compensation is part of preparing the state, not merely an improvement in count rate.

A primary example pumped a 3-mm beta-barium-borate crystal with a $351.1\mathrm{nm}$ argon-ion laser. Collecting the two intersections of its type-II emission cones selected paired output directions. Half-wave rotation and compensating birefringent crystals reduced temporal walk-off, and polarization analysers with avalanche photodiodes measured coincidence fringes. Varying analyser settings tested the correlations; accidentals, mode distinguishability and detector/collection assumptions limited interpretation. The reported visibility exceeded 97 percent and supported a Bell violation under the stated auxiliary assumptions, rather than a loophole-free claim.^[ref]

Other uses include preparing interfering photons for quantum-information experiments, correlation-based imaging, and pair-based detector calibration. All rely on specified spatial and spectral modes and coincidence statistics. A pump-blocked background run, delayed-coincidence estimate, gain dependence and polarization/temporal-overlap scans distinguish correlated pairs from fluorescence, detector dark counts and accidental coincidences. The useful resource is the controlled joint quantum state, not simply two bright output beams.

The stated annihilation conditions can be solved algebraically, even though (b) shows they are not ground-state conditions for the full printed Hamiltonian. Invert the canonical transformation:For $|{\psi }_{\lambda }⟩=\mathcal{N}{e}^{\lambda a^{†}{b}^{†}}|0,0⟩$, the commutators giveThus both $A$ and $B$ annihilate it precisely whenThe normalized two-mode squeezed vacuum isThe factorial from the exponential cancels those generated by the two creation operators. Normalization follows from a geometric series; the problem does not require its value, but it makes the subsequent partial trace explicit.

For comparison, the actual ground wave function of the printed stable Hamiltonian isIt follows directly from the two normal-mode Gaussian vacua. The proposed $|{\psi }_{\lambda }⟩$ is instead the exact ground state of the distinct pair-coupling Hamiltonian $\hslash \omega (a^{†}a+b^{†}b+1)+g(ab+a^{†}{b}^{†})$ under its stability condition. Omitting the remaining terms of $kxy$ silently would change the problem.