Draft:Displaced squeezed state

Operator representation

The general form of a displaced squeezed state for a quantum harmonic oscillator is given by

$|\zeta ,\alpha \rangle =D(\alpha )|\zeta \rangle =D(\alpha )S(\zeta )|0\rangle$

where $|0\rangle$ is the vacuum state, $D(\alpha )$ is the displacement operator and $S(\zeta )$ is the squeeze operator, given by

$D(\alpha )=\exp(\alpha {\hat {a}}^{\dagger }-\alpha ^{*}{\hat {a}})\qquad {\text{and}}\qquad S(\zeta )=\exp {\bigg [}{\frac {1}{2}}(\zeta ^{*}{\hat {a}}^{2}-\zeta {\hat {a}}^{\dagger 2}){\bigg ]}$

where $\alpha =|\alpha |e^{i\phi }$, and $\zeta =re^{i\theta }$. ${\hat {a}}$ and ${\hat {a}}^{\dagger }$ are annihilation and creation operators, respectively. For a quantum harmonic oscillator of angular frequency $\omega$, these operators are given by

${\hat {a}}^{\dagger }={\sqrt {\frac {m\omega }{2\hbar }}}\left(x-{\frac {ip}{m\omega }}\right)\qquad {\text{and}}\qquad {\hat {a}}={\sqrt {\frac {m\omega }{2\hbar }}}\left(x+{\frac {ip}{m\omega }}\right)$

For a real $\zeta$, where r is squeezing parameter), the uncertainty in $x$ and $p$ are given by

$(\Delta x)^{2}={\frac {\hbar }{2m\omega }}\mathrm {e} ^{-2\zeta }\qquad {\text{and}}\qquad (\Delta p)^{2}={\frac {m\hbar \omega }{2}}\mathrm {e} ^{2\zeta }$

Therefore, a squeezed coherent state saturates the Heisenberg uncertainty principle $\Delta x\Delta p={\frac {\hbar }{2}}$, with reduced uncertainty in one of its quadrature components and increased uncertainty in the other.

Some expectation values for displaced squeezed state are

$\langle \zeta ,\alpha |{\hat {a}}|\zeta ,\alpha \rangle =\alpha$

$\langle \zeta ,\alpha |{\hat {a}}^{2}|\zeta ,\alpha \rangle =\alpha ^{2}-e^{i\theta }cosh(r)sinh(r)$

$\langle \zeta ,\alpha |{\hat {a}}^{\dagger }{\hat {a}}|\zeta ,\alpha \rangle =|\alpha |^{2}+sinh^{2}(r)$

The general form of a squeezed coherent state for a quantum harmonic oscillator is given by

$|\alpha ,\zeta \rangle ={\hat {S}}(\zeta )|\alpha \rangle ={\hat {S}}(\zeta ){\hat {D}}(\alpha )|0\rangle$

Some expectation values for squeezed coherent states are

$\langle \alpha ,\zeta |{\hat {a}}|\alpha ,\zeta \rangle =\alpha cosh(r)-\alpha ^{*}e^{i\theta }sinh(r)$

$\langle \alpha ,\zeta |{\hat {a}}^{2}|\alpha ,\zeta \rangle =\alpha ^{2}cosh^{2}(r)+{\alpha ^{*}}^{2}e^{2i\theta }sinh^{2}(r)-(1+2{|\alpha |}^{2})e^{i\theta }cosh(r)sinh(r)$

$\langle \alpha ,\zeta |{\hat {a}}^{\dagger }{\hat {a}}|\alpha ,\zeta \rangle =|\alpha |^{2}cosh^{2}(r)+(1+{|\alpha |}^{2})sinh^{2}(r)-({\alpha }^{2}e^{-i\theta }+{\alpha ^{*}}^{2}e^{i\theta })cosh(r)sinh(r)$

Since ${\hat {S}}(\zeta )$ and ${\hat {D}}(\alpha )$ do not commute with each other,

${\hat {S}}(\zeta ){\hat {D}}(\alpha )\neq {\hat {D}}(\alpha ){\hat {S}}(\zeta )$

$|\alpha ,\zeta \rangle \neq |\zeta ,\alpha \rangle$

where ${\hat {D}}(\alpha ){\hat {S}}(\zeta )={\hat {S}}(\zeta ){\hat {S}}^{\dagger }(\zeta ){\hat {D}}(\alpha ){\hat {S}}(\zeta )={\hat {S}}(\zeta ){\hat {D}}(\gamma )$, with $\gamma =\alpha \cosh r+\alpha ^{*}e^{i\theta }\sinh r$