General multimode difference-squeezing

We consider a quantum process in a nonlinear medium in which a number of input modes interact to generate an output mode with a general difference-frequency. We introduce for the input modes a concept of general multimode difference-squeezing whose relationship with normal squeezing of the output mode is established. We then analyze the conditions for the input to be multimode difference-squeezed in dependence on the individual modal states. Finally, we study possible connection of the general multimode difference-squeezing with a symmetry group. 2000 Elsevier Science B.V. All rights reserved.

Multimode first-order squeezing ( seec,e.g,Refs,[6,7]) was also considered for which the ‘‘quadrature’’ operator is defined as a linear superposition ( i.e., asum ) of the modes’ operators

Single-mode higher-order squeezed states are meaningful as well. Their first version was suggested in[8]where higher-powered variances are dealt with instead of the conventional variance. Another version ofsingle-mode higher-order squeezing was defined in[9;10]where amplitude-squared squeezing was introduced. ofThis was afterward extended to amplitude-cubed[11]and amplitude-K-powered(see, e.g.[12;13]) .squeezing.A nice nontrivial unified single-mode higher-order squeezing operator[14]based on non-Hermitian realizations Athe relevant algebras’ operators by means of generalized multiboson operators[15]was also constructed

The so-called sum- and difference-squeezing were proposed for the first time in[16]for two modes which are in fact the simplest versions of multimode higher-order squeezing. These concepts have recently been generalized to include three modes for sum-[17] and difference-squeezing [18] as well as an arbitrary number of modes for sum-squeezing [19]. Sum- as well as difference-squeezing are multimode and, at the same time, higher-order since their underlying ‘‘quadrature’’ operator is defined in terms of a product (not a sum ) of themodes’ operators.

Like sum-squeezing, it is natural to generalize the two-mode [16] and three-mode [18]difference-squeezing to the most general multimode case. That is the purpose of this paper. In Section 2 we define the most general multimode difference-squeezing and study its nonclassical property. In Section 3 we establish the relation converting the input general multimode difference-squeezing to the output normal squeezing. Section 4

elucidates the condition for the input system to be multimode difference-squeezed in dependence on the modal squeezings. Possible formal identification between the general difference-squeezing operators and the generators of a symmetry group is made in Section 5. In the final section we contrast the general multimode sum-squeezing

[19] with difference-squeezing and conclude.

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