Minimum-uncertainty states for amplitude-squared squeezing

Normal squeezing is described in terms of variables which are linear in the field creation and annihilation operators. Amplitude-squared squeezing, on the other hand, is defined in terms of variables which are quadratic in these operators. As a result processes, such as second-harmonic generation, which couple one mode amplitude to the square of another convert this kind of squeezing into normal squeezing. In particular, the variables one considers are the hermitian part and one over i times the antihermitian part of the square of the annihilation operator. The variances of these observables obey an uncertainty relation. In order to better understand the properties of amplitude-squared squeezing it is useful to find the minimum uncertainty states corresponding to this uncertainty relation. These states are described by two parameters; the first,, gives the expectation of the square of the annihilation operator and the other λ, describes the amount of amplitude-squared squeezing. AH of these states have a mean amplitude of zero. These states for which =0 have unusual noise properties in that their Q functions have a four-fold rotational symmetry. We also find that a squeezed vacuum state is an amplitude-squared minimum uncertainty state.