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Such a density operator $\hat{\rho}$ match with a Wigner function in coherent space: $$W(\alpha,\alpha^{*})$$ $\alpha$ and $\alpha^{*}$ are $C-number$ ( $\alpha^{*}$ denotes complex conjugate of $\alpha$) corresponding to annihilation and creation operator $a$ and $a^\dagger$.

Or we may write Wigner function under coordinate space: $$W(x,p)$$ where $x$ and $p$ are also $C-number$ corresponding to position and momentum operator $\hat{x}$ and $\hat{p}$.

My question is, if I have a Wigner function under coordinate space $W(x,p)$, may I obtain $W(\alpha,\alpha^{*})$ just simply get through: $$\alpha→x+p$$ $$\alpha^*→x-p$$ or vice versa: $$x=(\alpha+\alpha^{*})/2$$ $$p=(\alpha-\alpha^{*})/2i$$ Is there such a simple relationship between them?

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  • $\begingroup$ WP. $\endgroup$ Commented Mar 19, 2022 at 11:54

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I think it suffices to do what you did. And generally if you want to include non-natural units or where everything isn't set to 1;

$$x=\sqrt{\frac{\hbar}{2m\omega}}(\alpha^* + \alpha),\, p=i\sqrt{\frac{\hbar m\omega}{2}}(\alpha^* - \alpha). $$

On a side note, the coherent representation is sometimes more useful because for example in circuit QED or cavity QED, since $|\alpha|^2$ is the average number of photons - this is immediately more useful since they can control "how many photons" to pump and explore or probe phase space experimentally.

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