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$$W(\alpha)=\frac{1}{\pi^2}\int e^{\lambda\alpha^*-\lambda^*\alpha} \operatorname{Tr}\left[ \hat{\rho}e^{\lambda\hat{a}^\dagger} e^{-\lambda^* \hat{a}} \right] e^{-\frac{|\lambda|^2}{2}} \, d^2\lambda. $$ (Ref: Eqn 3.136 on Page 67 in "Introductory quantum optics" by C. Gerry and P. Knight (2005))

My question is about $d^2\lambda$. Is it just $d(\operatorname{Re}[\lambda]) \, d(\operatorname{Im}[\lambda])$ or something else?

What are the techniques generally used to solve these integrals if they are something else?

sash
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Since $\lambda=x+ip$, $d\lambda^* \,d\lambda$ is basically $dx \, dp$ (as expected). Moreover, the factor $e^{-\lambda \lambda^*/2}$ is a Gaussian tail, which usually means integration by parts will come handy.

Unless we know more about $\rho$ there’s nothing else to say here. If $\rho$ is pure and a harmonic oscillator eigenstate $\vert n\rangle$, the integral is doable for any $n$ in terms of known special functions: you can find details in Schleich, Wolfgang P., Quantum optics in phase space (John Wiley & Sons, 2011).

Emilio Pisanty
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ZeroTheHero
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