# Photon number statistics of a displaced thermal state

What I am looking for is the possibility of detecting $n$ photon in a displaced thermal state

$\langle n|\rho_{dts}|n\rangle$

$\rho_{DTS} = D(\alpha)\rho_{TS} D(-\alpha)$

where $\rho_{TS}$ is a thermal state with mean photon number $B$ and $D(\alpha)$ is the displacement operator.

I have already found an analytical expression for the first 4 term:

$\langle n|\rho_{dts}|n\rangle = exp(-\alpha^2/(B+1))P_n$

Where $P_0$ to $P_3$ could be found in this post:

https://math.stackexchange.com/questions/2844741/try-to-guess-a-closed-form-expression

## 1 Answer

Assuming you are referring to single mode non-interacting oscillator.

$$\textbf{Hint :}$$ Define $$\mathbb{G}[\lambda_{}^{}]=\sum_{n=0}^{\infty}e_{}^{i\lambda n}\langle n|\rho_{}^{}| n\rangle=\sum_{n=0}^{\infty}\langle n|e_{}^{i\lambda a_{}^{\dagger}a_{}^{}}\rho_{}^{}| n\rangle=\textbf{Tr}[e_{}^{i \lambda a_{}^{\dagger}a_{}^{}} \rho_{}^{}]$$.

Now $$\mathbb{G}[\lambda_{}^{}]$$ can be easily evaluated by taking trace in coherent state basis for the choice $$\rho_{}^{}=\rho_{DTS}^{}$$. (you will encounter only gaussian integrals).

$$\langle n|\rho_{DTS}^{}|n\rangle$$ can be generated as Fourier expansion coefficients or by taylor expanding $$\mathbb{G}[-i\log(s)]$$ in $$s$$.

$$\textbf{Note :}$$ See this answer.

• Nice trick. I think the trace could also easily be calculated using the Wigner functions. Then the result could be easily generalized to photon number statistics of any Gaussian state. – L.Han Jul 10 '18 at 16:55