# Diagonalisation of quasi-thermal state

I have the following density operator $$\frac{1}{t \pi N} \int_{\mathbb{C}} \mathrm{d}^2\gamma \exp \left[ -\frac{|\gamma+r\alpha|^2}{t^2 N} \right] |{\gamma}\rangle\langle{\gamma}|,$$ where $$0\leq t,r\leq 1$$ are fixed, and $$\alpha \in \mathbb{C}$$ is also a constant. The integral runs over coherent states, and $$\mathrm{d}^2\gamma := \mathrm{d}\Re(\gamma)\mathrm{d}\Im(\gamma)$$ (where both real and imaginary parts run from $$-\infty$$ to $$+\infty$$ to span the whole of $$\mathbb{C}$$).

I would like to find the eigenstates of this operator. Although it is already diagonal, I am having trouble identifying the eigensolutions. I hope this is a very silly question and get a quick reply!

• Isn't this diagonal in the gamma basis already? – Pratik Rath Jun 15 at 13:04
• @PratikRath. Coherent states are not orthogonal, so they do not compose a basis (although they are complete). – march Jun 20 at 16:05