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Statistics of multiphoton emission

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Our last paper with Guillermo Díaz Camacho et al. on the dynamics of emission of multiphotons, in particular, of photon bundles (?!) involves a lot of complicated combinatoric formulas, such as the average time of detection for the $k$th photon from a $N$-photon Fock state that is spontaneously emitted with a radiative rate $\gamma_a$ and measured by a detector with bandwidth $\Gamma$: $$\begin{multline} \langle t_{k}^{(N)} \rangle = \\\frac{k \gamma_a^N}{P(N,N)} \left( \frac{\Gamma}{\Gamma_-} \right) ^{2N} {N\choose k} \sum_{k_1 + k_2 + k_3\atop = N-k} {N-k\choose k_1, k_2, k_3} \sum_{k_4 + \cdots + k_9\atop = k-1}{k-1 \choose k_4,k_5,k_6,k_7,k_8,k_9}\sum_{k_{10} + k_{11} = 2} {2\choose k_{10},k_{11}} \\ \frac{(-1)^{k_3 + k_6 + k_7 + k_8 + k_{11}}}{\gamma_a^{k_1 +k_4+k_7} \Gamma^{k_2+k_5+k_8} \left( \frac{\Gamma_+}{4} \right)^{k_3 + k_6 + k_9} ( \gamma_a (k_1 + k_7+k_{11}/2) +\Gamma( k_2 +k_8 +k_{10}/2) + \frac{\Gamma_+}{2} (k_3+k_9))^2 }\,.\tag{eq:tkN:label exists!} \end{multline}$$

Here I provide a Mathematica code to turn this general result into particular and limiting cases, and discuss briefly the whole thing. The formula is obtained by integration over the joint probability density function $\phi_\Gamma$ for emitting the photons at times $t_i$, which we show in this paper is given by: $$\phi_\Gamma(t_1,\dots,t_N)=(-1)^{N}N!\gamma_a^N\left(\frac{\Gamma}{\Gamma_-}\right)^{2N }\prod_{i=1}^{N}\left[{d\over dt_i}({e^{-\gamma_a t_i}\over\gamma_a}+{e^{-\Gamma t_i}\over\Gamma}-4{e^{-\Gamma_+ t_i/2}\over\Gamma_+})\right]{\mathbf{1}}_{[t_{i-1}, t_{i+1}[}(t_{i})$$ where $\Gamma_\pm\equiv\Gamma\pm\gamma_a$. When tracing out all the photons to keep only the $k$th of interest, through some clever math hackery, we arrive to the computation of $\int_0^\infty\left({e^{-\gamma_a t}\over\gamma_a}+{e^{-\Gamma t}\over\Gamma}-4{e^{-\Gamma_+ t/2}\over\Gamma_+}\right)^N\,dt$ which is easy to do thanks to the multinomial theorem, which is where all the mad coefficients come from. That's how one gets down to something like Eq. (1)

This is the formula for the average of the squared time of the $k$th photon from a $N$-photon bundle:

$$\begin{multline} \langle \big(t_k^{(N)}\big)^2\rangle = \frac{2\gamma_a^N}{P(N,N)}\left( \frac{\Gamma}{\Gamma_-} \right)^{2N}k {N\choose k} \sum_{k_1+k_2+k_3=N-k}{N-k\choose k_1, k_2, k_3}\\{}\times \sum_{k_4+\cdots+k_9=k-1}{k-1\choose k_4, k_5, k_6, k_7,k_8,k_9} \sum_{k_{10}+k_{11}=2}{2\choose k_{10}, k_{11}}\\{}\times \frac{(-1)^{k_3 + k_6 + k_7 + k_{8}+k_{11}}}{\gamma_a^{k_1+k_4+k_7} \Gamma^{k_2+k_5+k_8} \left( \frac{\Gamma_+}{4} \right)^{k_3 + k_6 +k_9} ( \gamma_a (k_1+k_7+k_{11}/2) +\Gamma(k_2+k_8 +k_{10}/2) + \frac{\Gamma_+}{2} (k_3+k_9))^3} \end{multline}$$