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


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

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


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


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


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

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 $$ \gamma=\alpha\cosh r + \alpha^* e^{i\theta} \sinh r $$