## Phase space interference and HusimiŽs function

### Douglas Mundarain and Jorge Stephany

The concept of interference in phase space proposed by Wheeler and Schleich some time ago *[Nature* **326**, 574 (1987)] provides an interesting tool to investigate the properties of quantum systems. The validity of the overlapping areas hypothesis has been demonstrated within the WKB approximation but in principle it is not restricted to this approximation. Also relevant is the relation of phase space interference with various the phase space distributions including Husimi Q function.
In **Phase space interference and the WKB approximation for squeezed number states.**
Squeezed number states for a single mode Hamiltonian are
investigated from two complementary points of view. Firstly the
more relevant features of their photon distribution are discussed
using the WKB wave functions. In particular the oscillations of
the distribution and the parity behavior are derived and compared
with the exact results. The accuracy of the approximation is
verified and it is shown that for high photon number it fails to
reproduce the true distribution. This is contrasted with the fact
that for moderate squeezing the WKB approximation gives the
analytical justification to the interpretation of the oscillations
as the result of the interference of areas with definite phases in
phase space. It is shown with a computation at high squeezing
using a modified prescription for the phase space representation
which is based on Wigner-Cohen distributions that the failure of
the WKB approximation does not invalidate the area overlap
picture.

In **Husimi's Q function and quantum interference in phase space** we discuss a phase space description of the photon number
distribution of non classical states which is based on Husimi's
Q function and does not rely in the WKB approximation.
We illustrate this approach using the examples of displaced number
states and two photon coherent states and show it to provide an
efficient method for computing and interpreting the photon number
distribution . This result is interesting in particular for the
two photon coherent states which, for high squeezing, have the
probabilities of even and odd photon numbers oscillating
independently.