# What is the effect of squeezing on the Husimi phase space representation or Q-function?

The effect of the squeezing operator \begin{equation} S = e^{- r (a^2 + a^{\dagger 2}) / 2} \end{equation} on a Wigner phase space representation or W-function of a system with density matrix $\rho$ \begin{equation} W(x, p) = \frac{1}{\pi \hbar} \int_{\mathbb{R}} dy e^{- 2 i p y / \hbar} \langle x + y | \rho | x - y \rangle \end{equation} is \begin{equation} S \circ W(x, p) = W(x e^r, p e^{-r}) \end{equation} (I am using the convention that $x = 2^{-1/2} (a + a^{\dagger})$ and $p = - i 2^{-1/2} (a - a^{\dagger})$). Is there a simple relation for the effect of the squeezing operator on the Husimi phase space representation or Q-function \begin{equation} Q(x, p) = \frac{2}{\pi} \int_{\mathbb{R}^2} du dv e^{- 2 (x - u)^2 - 2 (p - v)^2} W(u, v) , \end{equation} i.e., \begin{equation} S \circ Q(x, p) = ? \end{equation} More generally, I'd also like to know if there is a simple relation for the effect of squeezing on a generalised phase space representation or R-function (sometimes also called an $s$-parametrised W-function) \begin{equation} R(x, p, \tau) = \frac{1}{\pi \tau} \int_{\mathbb{R}} du dv e^{- \tau^{-1} (x - u)^2 - \tau^{-1} (p - v)^2} P(u, v) \end{equation} where $P(x, p)$ is the Glauber-Sudarshan phase space representation or P-function. $P(x, p)$ satisfies \begin{equation} \rho = \int_{\mathbb{C}} d\alpha P(\alpha) |\alpha\rangle \langle \alpha | \end{equation} where $|\alpha\rangle$ are the coherent states of $a$. The reason I ask is that I have developed some numerical algorithms that calculate a generalised phase space representation for certain states very accurately and I'd now like to see the effect of squeezing, preferably without having to perform numerical integration etc.

Isn't this question a bit ambiguous?

More directly, skipping the superfluous creating and annihilation operators, the symplectic dilation operator on phase-space variables is just $$S= e^{r(x\partial_x- p \partial_p)} ,$$ so that $$S W(x,p)= W(x e^r , p e^{-r} )$$ Further note $S\partial_x= e^{-r} \partial_x$ and $S \partial_p= e^r \partial_p$.

The Husimi function is the Weierstrass transform of the WF, $$Q(x,p)=T W(x,p),$$ where $$T\equiv e^{\frac{\hbar}{4} (\partial_x^2+\partial_p^2)} ,$$ eqn (122) of Ref 1, equivalent to the integral kernel you provide (in which you have implicitly chosen/absorbed $\hbar=1/2$.)

Thus, purely formally as you are asking, $$SQ(x,p)=Q(x e^r,p e^{-r})= e^{\frac{\hbar}{4} (e^{-2r}\partial_x^2+e^{2r}\partial_p^2)} W(xe^{r},pe^{-r}) .$$

I assume, however, you are asking the question because you are interested, instead, in the Husimi function of a squeezed state, (ref. 2 Chapter 12.2 and prob 12.1): that is, low-pass-filtering a squeezed WF, SW, with an unsqueezed Gaussian, that is leaving the Weierstrass transform operator T unsqueezed.

Your choices are dictated by the application you are interested in, for example, whether you are fussing with the characteristic star Husimi-Voros product or not, etc.

References:

1. Thomas L. Curtright, David B. Fairlie, & Cosmas K. Zachos, A Concise Treatise on Quantum Mechanics in Phase Space, World Scientific, 2014. The PDF file is available here.

2. Wolfgang P. Schleich, Quantum Optics in Phase Space, Wiley , 2011.

• Is there an analogous relation for wave functions? If $\psi(x) = \langle x|\psi \rangle$ then can I write $\langle x |\phi \rangle =\langle x| S(r) |\psi \rangle$ in terms of $\psi(x)$? For sure one can show directly that $|\phi(x)|^2 = e^r |\psi(x e^r)|^2$ but of course one still does not know how the phase of the squeezed wave function is modified. Equivalently, what is the action of the squeezing operator on quadrature eigenstates? Dec 12 '17 at 16:20
• Well, you just computed squeezing on $W(x,p=0)$, of course, by the very definition of the Wigner function. The phase information, and its concomitant squeezing is encoded in the p - dependence. That's the point of phase-spacery: to completely forget about wavefunctions, and practice clean living henceforth.... Standard treatises do Fock and coherent states, etc... Dec 12 '17 at 17:34