Phase space representations and coherent state representation of $\rho$, W, P and Q functions in quantum optics and related questions

Question

• What is the difference between phase space representations (with real x & p as variables) and coherent state representation (with complex $\alpha$ as a variable) of $\rho$, W, P and Q functions in quantum optics?

• How do first quantization and second quantization relate to different representations?

• Are these two representations equivalent?

• When do we use one or the other?

• How and where does the Fermionic or Bosonic nature of the things we are describing using these tools come in?

• Why do we need W, P and Q functions if they are derived from $\rho$?

Answer 1

With a quantum communication engineer background, I'll try to ask you and expand Cosmas's answer as best as possible. I'm sorry if I can't give you further physical insights.

The difference between the $x-p$ plane, with $x,p\in\mathbb{R}$, and the $\alpha$ plane, with $\alpha\in\mathbb{C}$ is just a change of variable (it is equivalent to see $\mathbb{C}$ as an $\mathbb{R}^2$ plane). You can try by yourself, e.g., starting from the definition of the symmetrically ordered characteristic function (the computations are long and boring): $$\chi(u,v)=\text{Tr}\left\{\rho e^{-i(uP+vX)}\right\}$$

And making a change of variable by using this relationship: $$a=\frac{\hat{x}+i\hat{p}}{\sqrt{2}}$$ Where $\hat{x}=\sqrt{\frac{\omega}{\hbar}}X$ and $\hat{p}=\frac{1}{\sqrt{\hbar\omega}}P$. You arrive at the form: $$\chi(\xi)=\text{Tr}\left\{\rho e^{\xi a^{\dagger}-\xi^*a}\right\}$$ Doing an analogue change of variable with the Wigner function (i.e., the Fourier transform of $\chi(u,v)$) leads you from $W(x,p)$ to $W(\alpha)$. This time you need to express $x,p$ as a function of $\alpha$ and $\alpha^*$. The same reasoning applies to the Glauber-Sudarshan $P$ function and Husimi $Q$ function.

The different phase-space representations ($W$, $P$ and $Q$) comes out from the so called operator-ordering problem. This problem (and its solution) is purely mathematical and it has no physical justification, as the three function represents exactly the same state. As a consequence of the operator ordering problem, you can use the different representations to compute, respectively, the simmetrically/normally/anti normally ordered moments of the state. This is a first reason in unsing different, but equivalent, phase space representations. Indeed, is easy to show that you can use the Wigner function to calculate the simmetrically ordered moments, i.e., $$\text{Tr}\left\{ \rho\mathcal{S}\left[a^n(a^{\dagger})^m\right] \right\}= \int\alpha^n(\alpha^*)^mW(\alpha)d^2\alpha$$ While the $P$ representation can be used to calculate the normally ordered moments, i.e., $$\text{Tr}\left\{ \rho (a^{\dagger})^ma^n \right\}= \int\alpha^n(\alpha^*)^mP(\alpha)d^2\alpha$$ Furthermore, the $P$ function is useful to represent a state in the overcomplete basis of coherent states, i.e., $$\rho = \int P(\alpha)|\alpha\rangle\langle\alpha|d^2\alpha$$

I can't answer the other more physical questions, but I hope to have clarified the most theoretical issues.