# Phase Space Representations in Quantum Optics

Graphical representations of quantum states of light are important to intuitively understand their noise properties. There are four different types of quantum states that are important to know when discussing squeezed light: The vacuum state, the coherent state, the displaced squeezed state and the squeezed vacuum state.

How are these states connected with each other? What is their impact on an experiment?

The picture below shows the four states in their phase space representation, spanned by the amplitude quadrature X and the phase quadrature Y. The red areas represent quantum noise projections of the corresponding Wigner functions.

The vacuum state is illustrated as a circle centered at the origin. Even when no other (classical) signal is apparent, the vacuum state couples into every path of an experiment Coupling occurs, for example, via empty ports of beam splitters and cannot be eliminated in a classical way. It always inevitably leads to noise in the experiment from the unused ports.

The coherent state approximates a conventional laser. The red arrow corresponds to the amplitude of the radiation. Thus, the higher the optical power of the laser, the longer this arrow becomes. The red circle illustrates the quantum uncertainty apparent in every conventional laser. Measuring the amplitude, the tip of the arrow would be found somewhere within the circle with a higher probability to be in the center than in the outer areas. Performing repetitive measurements, one would observe that the amplitude of the coherent state fluctuates.

The vacuum state can be squeezed, leading to a squeezed vacuum state. As a result, a reduced uncertainty along one axis, in this example the x-axis, is achieved. This is only possible at the expense of an increased uncertainty in the orthogonal axis, here the y-axis.

A squeezed vacuum state can be displaced, leading to a displaced squeezed state. In practice, the displacement happens by overlapping a squeezed vacuum state with a coherent state on a beamsplitter, e.g. with a 90/10 or 99/1 splitting ratio. The result is a state that has an amplitude (indicated by the red arrow) like a conventional laser but reduced noise along one axis. The picture shows a noise reduction along the x-axis and thus an amplitude squeezed state.

Compared to a coherent state from a conventional laser, this noise reduction allows for more precise measurements!