Editing Squeezed coherent state (section)

== Relation with the concept of quantum phase space ==
The concept of Quantum Phase Space (QPS) extends the notion of [[phase space]] from classical to [[Quantum mechanics|quantum physics]] by taking into account the [[uncertainty principle]]. The definition of the QPS is based on the introduction of joint momentum-coordinate quantum states denoted <math>|\langle z \rangle\rangle</math> which can be considered as some kind of squeezed coherent states. The expression of the wavefunction corresponding to a state <math>|\langle z \rangle\rangle</math>  in coordinate representation is<ref name=":1">R.T. Ranaivoson et al : "''Invariant quadratic operators associated with linear canonical transformations and their eigenstates''", [https://iopscience.iop.org/article/10.1088/2399-6528/ac8520 J. Phys. Commun. 6 095010], [[arxiv:2008.10602|arXiv:2008.10602 [quant-ph]]], (2022) </ref>

<math>\varphi(x)=\langle x|\langle z\rangle\rangle =\frac{e^{- \frac{B}{(\hbar)^2}(x-\langle x\rangle)^2+\frac{i}{\hbar}\langle p \rangle x}}{(2\pi A)^\tfrac{1}{4}}</math>

in which :

* <math>\hbar</math> is the reduced [[Planck constant]]
* <math>x</math> and <math>p</math> are respectively the eigenvalues (possible values) of the coordinate operator <math>X</math> and the momentum operator <math>P</math>
* <math>\langle x\rangle</math> , <math>\langle p\rangle</math>, <math>A</math> and <math>B</math> are respectively the mean values  and statistical variances of the coordinate and momentum corresponding to the quantum state  <math>|\langle z \rangle\rangle</math> itself

<math>\begin{cases} \langle x\rangle = \langle \langle z \rangle|X|\langle z\rangle\rangle \\ \langle p\rangle = \langle \langle z \rangle|P|\langle z \rangle \rangle \\ A=\langle \langle z\rangle|(X-\langle x\rangle)^2|\langle z \rangle \rangle \\ B =\langle \langle z \rangle|(P-\langle p\rangle)^2|\langle z \rangle\rangle \end{cases}</math>

A state <math>|\langle z \rangle\rangle</math>  saturates the uncertainty relation i.e. one has the following relation

<math>\sqrt{A} \sqrt{B}=\frac{\hbar}{2}</math>

It can be shown that a state <math>|\langle z \rangle\rangle</math> is an eigenstate of the operator <math>Z=P - \frac{2i}{\hbar}BX </math>. The corresponding eigenvalue equation is

<math>Z|\langle z \rangle\rangle =\langle z \rangle|\langle z \rangle\rangle</math>

with

<math>\langle z \rangle = \langle p \rangle - \frac{2i}{\hbar} B\langle x\rangle </math>

It was also shown that the multidimensional generalization of the states  <math>|\langle z \rangle\rangle</math> are the basic quantum states which corresponds to wavefunctions that are covariants under the action of the [[Group action|group]] formed by multidimensional [[Linear canonical transformation|Linear Canonical Transformations]].<ref>R. T. Ranaivoson et al: "''Linear Canonical Transformations in relativistic quantum physics''", [https://iopscience.iop.org/article/10.1088/1402-4896/abeba5 Phys. Scr. 96, 065204], [[arxiv:1804.10053|arXiv:1804.10053 [quant-ph]]], (2021) </ref> 

The quantum phase space (QPS) is defined as the set <math>\{\langle z\rangle\}</math> of all possible values of <math>\langle z\rangle</math> , or equivalently as the set <math>\{(\langle p \rangle ,\langle x\rangle)\}</math> of possible values of the pair <math>(\langle p \rangle ,\langle x\rangle)</math>,  for a given value of the momentum statistical variance <math>B</math>.<ref name=":1" /> It follows from this definition that the structure of the quantum phase space depends explicitly on the value of the momentum statistical variance. It is this explicit dependence that makes this definition naturally compatible with the uncertainty principle. It can also be remarked here that, at thermodynamic equilibrium, the momentum statistical variance <math>B</math> can be related to thermodynamics parameters like temperature, pressure and volume shape and size.<ref name=":3">R. H. M. Ravelonjato et al (2023) "''Quantum and Relativistic corrections to Maxwell-Boltzmann ideal gas model from a Quantum Phase Space approach''":  [https://link.springer.com/article/10.1007/s10701-023-00727-5 ''Found Phys'' '''53''', 88], [[arxiv:2302.13973|arXiv:2302.13973 [cond-mat.stat-mech]]]</ref> At the classical limit, when the momentum and coordinate statistical variances are taken to be equal to zero (ignoring the uncertainty principle), the quantum phase space as defined previously is reduced to the classical phase space.

There are more generalized squeezed coherent states, denoted <math>|n,\langle z \rangle\rangle</math> with <math>n</math> a positive integer, that are related to the concept of QPS and which do not saturate the uncertainty relation for <math>n>0</math>. These states can be deduced from to the states <math>|\langle z \rangle\rangle</math> using the following relation<ref name=":1" />

<math>|n,\langle z \rangle\rangle =\frac{1}{\sqrt{n!}}[\frac{-i(Z-\langle z\rangle)^\dagger}{2\sqrt{B}}]^n |\langle z \rangle\rangle</math>

The coordinate and momentum statistical variances, denoted respectively <math>A_n</math> and <math>B_n</math>, corresponding to a state <math>|n,\langle z \rangle\rangle</math> are

<math>\begin{cases}A_n =\langle n, \langle z \rangle|(X-\langle x \rangle)^2|n, \langle z \rangle\rangle =(2n+1)A \\B_n =\langle n, \langle z \rangle|(P-\langle p\rangle)^2|n,\langle z \rangle \rangle=(2n+1)B  \end{cases}</math> 

We then have the following relation

<math>\sqrt{A_n}\sqrt{B_n}=(2n+1)\frac{\hbar}{2}\geq \frac{\hbar}{2}</math>

This relation shows that a state <math>|n,\langle z \rangle\rangle</math>  does not saturate the uncertainty relation for <math>n>0</math> as said before.