Editing Squeezed coherent state (section)

== Operator representation ==
The general form of a '''squeezed coherent state''' for a quantum harmonic oscillator is given by

:<math> |\alpha,\zeta\rangle = \hat{S}(\zeta)|\alpha\rangle = \hat{S}(\zeta) \hat{D}(\alpha)|0\rangle  </math>

where <math>|0\rangle</math> is the [[vacuum state]], <math>D(\alpha)</math> is the [[displacement operator]] and <math>S(\zeta)</math> is the [[squeeze operator]], given by

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

where <math>\hat a</math> and <math>\hat a^\dagger</math> are annihilation and creation operators, respectively. For a [[quantum harmonic oscillator]] of angular frequency <math>\omega</math>, these operators are given by

:<math>\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)</math>

For a real <math>\zeta</math>, (note that <math>\zeta = r e^{2 i \theta}</math>,<ref>Walls, D.F. and G. J. Milburn, Quantum Optics. </ref> where ''r'' is squeezing parameter),{{clarify|reason=So this is r?|date=September 2016}} the uncertainty in <math>x</math> and <math>p</math> are given by

:<math>(\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}</math>

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

Some expectation values for squeezed coherent states are

:<math> \langle\alpha,\zeta | \hat a | \alpha,\zeta\rangle = \alpha \cosh(r) - \alpha^{*}e^{i\theta} \sinh(r) </math>

:<math> \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)   </math>

:<math> \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) </math>

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

:<math> |\zeta,\alpha\rangle = \hat{D}(\alpha)|\zeta\rangle = \hat{D}(\alpha) \hat{S}(\zeta)|0\rangle  </math>

Some expectation values for displaced squeezed state are

:<math> \langle\zeta,\alpha | \hat a | \zeta,\alpha\rangle = \alpha  </math>

:<math> \langle\zeta,\alpha | {\hat{a}}^2 | \zeta,\alpha\rangle = \alpha ^{2} - e^{i\theta} \cosh (r) \sinh (r)  </math>

:<math> \langle\zeta,\alpha | {\hat{a}}^{\dagger}\hat{a} | \zeta,\alpha\rangle = |\alpha|^2 + \sinh^2 (r) </math>

Since <math> \hat{S}(\zeta) </math> and <math> \hat{D}(\alpha)</math> do not commute with each other, 

: <math>\hat{S}(\zeta) \hat{D}(\alpha) \neq \hat{D}(\alpha) \hat{S}(\zeta)</math>

:<math> | \alpha, \zeta \rangle \neq | \zeta, \alpha \rangle </math>

where <math> \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)</math>, with <math> \gamma=\alpha\cosh r + \alpha^* e^{i\theta} \sinh r </math> <ref> M. M. Nieto and D. Truax (1995), {{cite journal|title=Holstein‐Primakoff/Bogoliubov Transformations and the Multiboson System|year=1997 |doi=10.1002/prop.2190450204|arxiv=quant-ph/9506025|last1=Nieto |first1=Michael Martin |last2=Truax |first2=D. Rodney |journal=Fortschritte der Physik/Progress of Physics |volume=45 |issue=2 |pages=145–156 |s2cid=14213781 }} Eqn (15). Note that in this reference, the definition of the squeeze operator (eqn. 12) differs by a minus sign inside the exponential, therefore the expression of <math>\gamma</math> is modified accordingly (<math>\theta \rightarrow \theta+\pi</math>). </ref>