Abstract
We present a complete analysis of variance matrices and quadrature squeezing 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,openR) of linear canonical transformations on n pairs of canonical variables. We exploit the transformation properties of variance (noise) matrices under symplectic transformations to express the uncertainty-principle restrictions 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 distinguish between photon-number-conserving passive linear optical systems and active ones. The former correspond to elements in the maximal compact U(n) subgroup of Sp(2n,openR), the latter to noncompact elements outside U(n). Based on this distinction, we motivate and state a U(n)-invariant squeezing criterion applicable to any state of an n-mode system, and explore alternative ways of expressing it. The set of all possible quantum-mechanical variance matrices is shown to contain several interesting 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 matrices in these subfamilies, as well as general ones, and their squeezing nature, are established. It is shown that all conceivable variance matrices can be generated through squeezed thermal states of the n-mode system and their symplectic transforms. Our formulas are developed in both the real and the complex forms for variance matrices, and ways to pass between them are given.
Highlights
Squeezed states of the radiation field are distinctly nonclassical in nature [l]
We examine the question of canonical forms for variance matrices in the context of our squeezing criterion
We have presented in this paper a comprehensive analysis of variance matrices and squeezing properties of arbitrary states, pure or mixed, of n-mode quantum systems
Summary
Squeezed states of the radiation field are distinctly nonclassical in nature [l]. Over the past decade their study has developed into a major area of quantum optics [2]. We present a comprehensive analysis of the properties of variance matrices for states of such systems Such matrices are real, symmetric, 2n dimensional, and positive definite. I1 we address the following basic question: Given a 2n X 2n real symmetric positive-definite matrix, how do we test whether it qualifies to be the variance matrix of some physically realizable state of the (quantum) n-mode system? The role of U(1) in the single-mode squeezing group Sp(2,R) is played by the U ( n ) - K ( n ) subgroup of Sp(2n,R) in the n-mode case This subgroup consists of all those phase-space rotations which are canonical transformations. We take care to state and explain them clearly at first encounter
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