# Mapping between quadrature squeezing and photon-number squeezing

It is possible to get quadrature squeezed light where on one quadrature the uncertainty is less than the of a coherent state while in the other one, orthogonal to it, it is larger. Is it possible, by doing an appropriate angle transformation, to go from a quadrature squeezed state to a photon-number squeezed state (sub-Poissonian distribution)?

• Could you clarify what you mean by photon-number squeezed state? Squeezed states can have sub-Poissonian statistics, and photon-number states can be squeezed but those are two quite different things, and the former is still squeezed along one quadrature (in a rotating frame). Sep 6, 2023 at 9:45
• @fulis I thought photon-number squeezing is synonymous with sub-Poissonian statistics (it is light where the photon number fluctuations are smaller than in Poisson distribution) Sep 13, 2023 at 7:19
• That type of state is commonly referred to as amplitude squeezed light (the complementary state is phase squeezed light). Sep 13, 2023 at 9:24

Yes. See the reference:

Squeezed States and Sub-Poissonian Photon Statistics L. Mandel Phys. Rev. Lett. 49, 136 – Published 12 July 1982

• Thank you. Can you please elaborate how the article explains this transformation? Sep 13, 2023 at 7:20

Yes it is.

Warning: I’m a theoretician, so everything I say below about something being “easy” is to be taken with a lot of salt

However, I think that in practice, doing a rotation is not a good idea, since the only “easy” rotation is a dephasing, which is a rotation around the origin and does not change the phase/amplitude nature of the squeezing. However, another “not too hard to implement” transformation is a displacement.It is usually closely approximated through interference with a bright coherent state over an almost transparent beamsplitter. Tuning carefully the amplitude of the coherent beam and its relative phase with the squeezed state such that the center of the latter is moved by 90% in phase space, converting the pahse-squeezing into photon number squeezing.

In absolute terms in the phase space, the squeezing direction has not changed, but since the phase of the mean complex amplitude has changed, the direction of what is called photon-number amplitude has changed, which does the trick

• This description is opaque and doesn’t really explain anything
– JQK
Sep 2, 2023 at 12:50