Abstract
After developing the concept of displaced squeezed vacuum states in the non-unitary approach and establishing the connection to the unitary approach we calculate their quasiprobabilities and expectation values in general form. Then we consider the displacement of the squeezed vacuum states and calculate their photon statistics and their quasiprobabilities. The expectation values of the displaced states are related to the expectation values of the undisplaced states and are calculated for some simplest cases which are sufficient to discuss their categorization as sub-Poissonian and super-Poissonian statistics. A large set of these states do not belong to sub- or to super-Poissonian states but are also not Poissonian states. We illustrate in examples their photon distributions. This shows that the notions of sub- and of super-Poissonian statistics and their use for the definition of nonclassicality of states are problematic. In Appendix A we present the most important relations for SU (1,1) treatment of squeezing and the disentanglement of their operators. Some initial members of sequences of expectation values for squeezed vacuum states are collected in Appendix E.
Highlights
The expectation values of the displaced states are related to the expectation values of the undisplaced states and are calculated for some simplest cases which are sufficient to discuss their categorization as sub-Poissonian and super-Poissonian statistics
What is the same, the displaced squeezed vacuum states belong to the most interesting states in quantum optics for which, practically, all interesting parameters and quasiprobabilities may be calculated in closed exact way
All minimum uncertainty states belong to the squeezed coherent states and some aspects of these states were already considered in the early years of the development of quantum mechanics not under this name, for example, by Schrödinger [1], Pauli [2] and Louisell [3]
Summary
In the narrow sense the squeezing operations form together with rotations in a plane (the two-dimensional phase plane) the Lie group SU (1,1) with 3 real parameters This Lie group possesses different realizations in quantum optics of a single mode and a basic nontrivial realization in a two-mode system. The squeezed coherent states are well appropriate to demonstrate some problems of the distinction of sub- and super-Poissonian photon statistics because the whole set of these states can be not assigned to only one of these two kinds of statistics and it requires substantial efforts to find out to which of these statistics it belongs in a special case.
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