# Partition Function for the Jaynes-Cummings Hamiltonian

I was wondering if anyone here has calculated before the partition function for the Jaynes-Cummings Hamiltonian: $H_{JC}=\omega_0 (a^\dagger a + \sigma^+ \sigma^-)+g (a \sigma^+ +a^\dagger \sigma^-)$ (Here I have assumed resonance and $a,a^\dagger$ are bosonic operators and $\sigma^-,\sigma^+$ are ladder operators for a spin one half particle) For the Hamiltonian above the energy of the ground state is zero and corresponds to 0 excitations in the harmonic oscillator and the spin being down. The excited eigenenergies come in pairs and are given by: $\omega_{n\pm}=n \omega_0 \pm \sqrt{n}g$. I am interested in knowing the partition function: $\mathcal{Z}=\text{tr}( \exp(-\beta H_{JC} ))=1+2\sum_{n=1}^\infty\exp(-\beta \omega_0 n )\cosh(\beta g \sqrt{n})$ I tried Mathematica to get an analytic expression for the above sum and it did not work. Any thoughts on whether the summation can be expressed in terms of some special function or how to calculate it numerically in an efficient and reliable way?

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