Squeezed vacuum state

Question

From:

Loudon, Rodney. The quantum theory of light. OUP Oxford, 2000.

Consider the single-mode quadrature-squeezed vacuum state defined by

$ | \zeta \rangle = \hat{S} (\zeta) | 0 \rangle $

where the squeeze operator is

$ \hat{S} (\zeta) = \text{exp} ( \frac{1}{2} \zeta^* \hat{a}^2 - \frac{1}{2} \zeta (\hat{a}^{\dagger})^2)$

where $\hat{a} $ and $ \hat{a}^{\dagger}$ are the destruction and creation operator for quantum harmonic oscillator

and $\zeta$ is the complex squeeze parameter

$\zeta = s e^{i \theta}$

Define the operators $\hat{X}$ and $ \hat{Y}$ like

$ \hat{X} = \frac{1}{2} ( \hat{a} + \hat{a}^{\dagger})$ and $ \hat{Y} = \frac{1}{2}\text{i}( \hat{a}^{\dagger} - \hat{a} )$

we can verify that

$ \langle \zeta | \hat{X} | \zeta \rangle $ = $ \langle \zeta | \hat{Y} | \zeta \rangle $ = 0

and the variances

$ (\Delta X )^2 = \frac{1}{4} [ e^{2s} \text{sin}^2(\frac{1}{2} \theta) + e^{-2 s} \text{cos}^2(\frac{1}{2} \theta) ] $

$ (\Delta Y )^2 = \frac{1}{4} [ e^{2s} \text{cos}^2(\frac{1}{2} \theta) + e^{-2 s} \text{sen}^2(\frac{1}{2} \theta) ] $

now he shows a representation of the quadrature expectation values

Question:

I am not understanding how he draws that ellipse and how he calculates the length of the axis. This is a rotated ellipse and I know the relationship with an ellipse with the axis parallel to the cartesian axis. But I still don't understand how he creates it.

Answer 1

The ellipse he draws is a cartoon of the Wigner function of the state, which is not discussed in the book. The Wigner function of a squeezed vacuum state is: $$ W(x,y) \propto \mathrm{exp}\big[-e^{2\zeta}x^2 - e^{-2\zeta}y^2\big]. $$ In other words a 2D Gaussian function, that is squeezed along one axis and stretched along another. For a detailed discussion on Wigner functions, including squeezed states, take a look at the book by Ulf Leonhardt.