Editing Coherent state (section)

==Thermal coherent state==
A single mode thermal coherent state<ref>{{cite journal | last1=Oz-Vogt | first1=J. | last2=Mann | first2=A. | last3=Revzen | first3=M. | title=Thermal Coherent States and Thermal Squeezed States | journal=Journal of Modern Optics | publisher=Informa UK Limited | volume=38 | issue=12 | year=1991 | issn=0950-0340 | doi=10.1080/09500349114552501 | pages=2339–2347| bibcode=1991JMOp...38.2339O }}</ref> is produced by displacing a thermal mixed state in [[phase space]], in direct analogy to the displacement of the vacuum state in view of generating a coherent state. The [[density matrix]] of a coherent thermal state  in operator representation reads

:<math>
\rho(\alpha, \beta)=\frac{1}{Z}D(\alpha)e^{-\hbar \beta\omega  a^{\dagger} a}D^{\dagger}(\alpha),
</math>

where <math>D(\alpha)</math> is the [[displacement operator]], which generates the coherent state <math>D(\alpha)|0\rangle=|\alpha\rangle</math> with complex amplitude <math>\alpha</math>, and <math>\beta=1/(k_B T)</math> . The [[partition function (quantum field theory)|partition function]] is equal to

:<math>
Z=\text{tr}\left\{ \displaystyle e^{-\hbar \beta\omega  a^{\dagger} a}\right\}=\sum_{n=0}^{\infty}e^{-n\beta \hbar\omega}=\frac{1}{1-e^{-\hbar\beta\omega}}.
</math>

Using the expansion of the identity operator in [[Fock states]], <math>I\equiv \sum_{n=0}^{\infty}|n\rangle\langle n|</math>, the [[density operator]] definition can be expressed in the following form

:<math>
\rho(\alpha, \beta)= \frac{1}{Z}\sum_{n=0}^{\infty}e^{-n\hbar\beta\omega} D(\alpha)|n\rangle\langle n| D^{\dagger}(\alpha)=\frac{1}{Z}\sum_{n=0}^{\infty}e^{-n\hbar\beta\omega}|\alpha,n\rangle\langle \alpha,n|,
</math>

where <math>|\alpha,n\rangle</math> stands for the displaced [[Fock state]]. We remark that if temperature goes to zero we have

:<math>
\lim_{\beta\to\infty}\rho(\alpha,\beta)=\lim_{\beta\to\infty}\sum_{n=0}^{\infty}e^{-n\hbar\beta\omega} (1-e^{-\hbar\beta\omega})|\alpha,n\rangle\langle \alpha,n|=\sum_{n=0}^{\infty} \delta_{n,0}|\alpha,n\rangle\langle\alpha,n|=|\alpha,0\rangle\langle\alpha,0|,
</math>

which is the [[density matrix]] for a coherent state. The average number of [[photons]] in that state can be calculated as below

:<math>
\langle n\rangle =\text{Tr}\{\rho a^{\dagger}a\}=\frac{1}{Z}\text{Tr}\{D^{\dagger}(\alpha)a^{\dagger}D({\alpha})D^{\dagger}(\alpha) a D(\alpha) e^{-\beta\hbar\omega a^{\dagger}a}\}=\frac{1}{Z}\text{Tr}\{(a^{\dagger} + \alpha^{*})(a + \alpha)e^{-\beta\hbar\omega a^{\dagger}a}\}=</math>

:<math>
=|\alpha|^2\frac{1}{Z}\text{Tr}\{e^{-\beta\hbar\omega a^{\dagger}a} \} + \frac{1}{Z}\text{Tr}\{a^{\dagger}a e^{-\beta\hbar\omega a^{\dagger}a}\}=|\alpha|^2 + \frac{1}{Z} \sum_{n=0}^{\infty}ne^{-n\beta\hbar\omega},
</math>

where for the last term we can write

:<math>
\sum_{n=0}^{\infty}ne^{-n\beta\hbar\omega}=-\frac{\partial}{\partial (\beta\hbar\omega)} \left( \sum_{n=0}^{\infty}e^{-n\beta\hbar\omega}\right)=\frac{e^{-\beta\hbar\omega}}{(1-e^{-\beta\hbar\omega})^2}.
</math>

As a result, we find

:<math>
\langle n\rangle=|\alpha|^2 +\langle n\rangle_{\text{th}},
</math>

where <math>\langle n\rangle_{\text{th}}</math> is the average of the [[photon]] number calculated with respect to the thermal state. Here we have defined, for ease of notation,

:<math>
\langle O\rangle_{\text{th}}=\frac{1}{Z}\text{tr}\{e^{-\beta\hbar\omega a^{\dagger}a}O\},
</math>

and we write explicitly

:<math>
\langle n\rangle_{\text{th}}=\frac{1}{e^{\beta\hbar\omega}-1}.
</math>

In the limit <math>\beta \to \infty</math> we obtain <math>\langle n\rangle=|\alpha|^2</math>, which is consistent with the expression for the [[density matrix]] operator at zero temperature. Likewise, the photon number [[variance]] can be evaluated as

:<math>
  \sigma^2=\langle n^2\rangle-\langle n\rangle^2=\sigma_{\text{th}}^2+|\alpha|^2\left(1+2\langle a^{\dagger}a\rangle_{\text{th}}\right),
</math>
with <math>\sigma_{\text{th}}^2=\langle n^2\rangle_{\text{th}}-\langle n\rangle_{\text{th}}^2</math>. We deduce that the second moment cannot be uncoupled to the thermal and the quantum distribution moments, unlike the average value (first moment). In that sense, the photon statistics of the displaced thermal state is not described by the sum of the [[Poisson statistics]] and the [[Boltzmann statistics]]. The distribution of the initial thermal state in phase space broadens as a result of the coherent displacement.