Non-classical states of light with smooth 𝑃-function

In quantum optics, the most fundamental criterion to judge the non- classicality of a quantum state of light is in terms of the Glauber- Sudarshan 𝑃 -function. If the 𝑃 -function of a state is not a valid prob- ability density, e.g. not a positive semi-definite function, the state is considered non-classical. However, most known non-classical states have a corresponding 𝑃 -function which is highly irregular. This renders working with them difficult and direct experimental reconstruction im- possible.
Here we introduce a new class of non-classical states with regular smooth 𝑃 -functions by "puncturing" a classical 𝑃 -function with nar- row negative peaks. We analytically proof their existence and deter- mine parameter ranges where the constructed states are physical, as well as the regimes yielding anti-bunching of light. To conclude, we present some possible experimental realizations of punctured states.