Creation and annihilation operators

The bosonic ladder operators satisfy

For the quantum harmonic oscillator with $a_{-}=P-im\omega X$, $a_{+}=a_{-}^{\ast }$ and $\gamma =2m\hslash \omega$, factorization gives $a_{+}{a}_{-}=2m(H-E_0)$, where $E_0=\hslash \omega /2$. For an energy eigenstate $\psi$ at $E_n=E_0+n\hslash \omega$,The first identity shows that lowering to the ground state is injective on the $E_n$ eigenspace; the second shows how to invert it by raising. If the ground eigenspace is one-dimensional, $(a_{-}{)}^n\psi =c{\psi }_0$ and therefore $\psi =c(a_{+}{)}^n{\psi }_0/({\gamma }^{n}n!)$. Thus every level is also nondegenerate. This proof explicitly rules out a second independent energy eigenstate at an excited level rather than merely constructing one ladder of states.

For real coefficients in $a_{\pm }=c_{\pm }X+i{d}_{\pm }P$, imposing $a_{-}^{†}=a_{+}$, $[a_{-},a_{+}]=I$ and $H=\hslash \omega (a_{+}{a}_{-}+\lambda I)$ givesIndeed the adjoint operator condition makes the two $c$ coefficients equal and the two $d$ coefficients opposite. The canonical commutation relation gives $2\hslash c_{-}{d}_{-}=1$, and matching the coefficients of $X^2$ and $P^2$ fixes their magnitudes. Both signs change together. The conventional creation and annihilation operators choose $\epsilon =1$; the other sign is an equivalent phase convention for the number states.

An annihilation operator lowers the occupation number operator eigenvalue by one and annihilates the Fock vacuum.

A creation operator raises the occupation number operator eigenvalue of a quantum harmonic oscillator by one. Repeated application builds Fock states.

For a real squeezing parameter $\gamma$, the single-mode squeeze operator isIts exponent is skew-adjoint, so $S(\gamma )$ is a unitary operator, and it implements the Bogoliubov transformation

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.

The number operator $N=A^{†}A$ satisfiesIt is self-adjoint and positive semidefinite.

For a bosonic mode satisfying the canonical commutation relation $[a,a^{†}]=1$, the number operator $n=a^{†}a$ obeys $n^2=a^{†}a{a}^{†}a=(a^{†}{)}^2{a}^2+n$. Thus a two-body on-site interaction is proportional to $n(n-1)$, with no self-interaction. In a fixed-number two-mode condensate with probability $q$ of occupying this mode, $⟨n(n-1)⟩=N(N-1)q^2$ by the binomial distribution.

For a hopping operator $X$ with $[\Delta ,X]=2X$, spectral functions obey $Xf(\Delta )=f(\Delta -2)X$. The coefficient on the left of the hop is evaluated on the final state. This ordering determines the shifted energy denominator in virtual-hopping calculations.

For a single electromagnetic field mode, the eigenvalues of the number operator count its photons.

A number state is a normalized eigenstate of the number operator:For a single quantum harmonic oscillator, it is also an energy eigenstate with energy $\hslash \omega (n+1/2)$.

If $N|\nu ⟩=\nu |\nu ⟩$, then $A$ lowers the eigenvalue by one and $\Vert A|\nu ⟩{\Vert }^2=\nu \Vert \nu ⟩{\Vert }^2$. Positivity forces the lowering ladder to terminate at eigenvalue zero, so every eigenvalue is a nonnegative integer.