Squeezed vacuum state

The squeezed vacuum is the zero-displacement squeezed coherent state $|\gamma ⟩=S(\gamma )|0⟩$. Its position and momentum variances are rescaled in opposite directions,so it remains an minimum-uncertainty state with $\Delta X\Delta P=\hslash /2$.

A real centered Gaussian wave packet with initial position variance $1/(2\alpha )$ and momentum variance ${\hslash }^2\alpha /2$ evolves under a quantum harmonic oscillator byThe covariance matrix obeys $\Sigma (t)=M\Sigma (0)M^{\mathsf{T}}$ and $\det \Sigma ={\hslash }^2/4$. Thus the Robertson-Schrödinger uncertainty relation stays saturated, although the position-momentum variance product can exceed ${\hslash }^2/4$ because of nonzero quantum covariance. The widths are stationary only for $\alpha =m\omega /\hslash$; otherwise they breathe with frequency $2\omega$.

photon addition to an optical squeezed vacuum gives $a^{†}S|0⟩=\cosh rS|1⟩$; subtraction gives $aS|0⟩=-\sinh rS|1⟩$. After successful-event normalization these are the same state up to phase, with mean occupation $1+3{\sinh }^{2}r$ and odd photon parity. At zero squeezing subtraction never succeeds; the normalized limiting state is not an output at that exact parameter value.

A single optical mode in a squeezed vacuum state. In principal optical quadratures, its variances are $e^{-2r}/4$ and $e^{2r}/4$, while its mean photon number is ${\sinh }^{2}r$. Its mean field vanishes. A degenerate optical parametric amplifier generates the state by pair creation with phase fixed by the pump. Loss admixes unsqueezed vacuum and limits the measured variance reduction.

The single-mode squeeze operator changes occupation in pairs. Consequently an optical squeezed vacuum has only even photon numbers, with mean $s={\sinh }^{2}r$ and variance $2s(s+1)$. Its large number fluctuations are compatible with quadrature squeezing: they concern different observables.

Here $s={\sinh }^{2}r$. The pair moment is $⟨N(N-1)⟩=3s^2+s$. These expressions follow from the single-mode Bogoliubov transformation or by differentiating the even-occupation probability generating function $G(t)=[\cosh r\sqrt{1-t^2{\tanh }^{2}r}{]}^{-1}$.

For an invertible finite-mode Bogoliubov transformation $a=A{a}^{\prime }+B{a}^{\prime }{}^{†}$, the symmetric squeezing matrix is $M=-A^{-1}B$. The canonical commutation relations make its singular values strictly less than one. The normalized state has $|C|=\det (I-M{M}^{†}{)}^{1/4}$. Infinitely many modes additionally require bosonic mode mixing implementability.

Pair creation in two bosonic modes gives the normalized state displayed. The two occupations coincide in each term, while either mode alone has a geometric number distribution. At small $r$, the leading nonvacuum term is a single pair. This optical construction does not require particle-antiparticle modes.

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.

Tracing out one mode leaves geometric occupation probabilities with $q=|\lambda {|}^2$. In an oscillator basis of frequency $\omega$, they are Boltzmann weights with $k_{B}T=\hslash \omega /\ln (1/q)$. The joint state can be pure while each reduced state is mixed.

For a pair of distinct bosonic particle and antiparticle modes, the normalized state obeying $(a+q{b}^{†})|\psi ⟩=(b+q{a}^{†})|\psi ⟩=0$ is $\sqrt{1-|q{|}^2}{\sum }_{n\ge 0}(-q{)}^n|n,n⟩$, for $|q|<1$. It contains equal occupations and is generated by a pair-creation exponential. Infinite products require a separate implementability or regulator check.

The pair count of a two-mode squeezed vacuum has $P(n)=(1-r)r^n$, where $r=|q{|}^2<1$. Its mean is $r/(1-r)$ and its variance is $r/(1-r{)}^2$. Equal mode occupations follow from pair creation, not from independent particle and antiparticle draws.