QUANTUM-NOISE MATRIX FOR MULTIMODE SYSTEMS - U(N) INVARIANCE, SQUEEZING, AND NORMAL FORMS

We present a complete analysis of variance matrices and quadrature squ eezing for arbitrary states of quantum systems with any finite number of degrees of freedom. Basic to our analysis is the recognition of the crucial role played by the real symplectic group Sp(2n, R) of linear canonical transformations on n pairs of canonical variables. We exploi t the transformation properties of variance (noise) matrices under sym plectic transformations to express the uncertainty-principle restricti ons on a general variance matrix in several equivalent forms, each of which is manifestly symplectic invariant. These restrictions go beyond the classically adequate reality, symmetry, and positivity conditions . Towards developing a squeezing criterion for n-mode systems, we dist inguish between photon-number-conserving passive linear optical system s and active ones. The former correspond to elements in the maximal co mpact U(n) subgroup of Sp(2n,R), the latter to noncompact elements out side U(n). Based on this distinction, we motivate and state a U(n)-inv ariant squeezing criterion applicable to any state of an n-mode system , and explore alternative ways of expressing it. The set of all possib le quantum-mechanical variance matrices is shown to contain several in teresting subsets or subfamilies, whose definitions are related to the fact that a general variance matrix is not diagonalizable within U(n) . Definitions, characterizations, and canonical forms for variance mat rices in these subfamilies, as well as general ones, and their squeezi ng nature, are established. It is shown that all conceivable variance matrices can be generated through squeezed thermal states of the n-mod e system and their symplectic transforms. Our formulas are developed i n both the real and the complex forms for variance matrices, and ways to pass between them are given.