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International Journal of Modern Physics B

TWO-MODE GAUSSIAN DENSITY MATRICES AND SQUEEZING OF PHOTONS

Robert R. Tucci

Published May 20, 1992• 2 citations
Publisher

Abstract

In this paper, we generalize to 2-mode states the 1-mode state results obtained in a previous paper. We study 2-mode Gaussian density matrices (i.e., density matrices of the form: exponential of a quadratic polynomial in the creation and annihilation operators for the two modes). We find a linear transformation which maps the two annihilation operators, one for each mode, into two new annihilation operators that are uncorrelated and unsqueezed. This allows us to express the density matrix as a product of two 1mode density matrices. We find general conditions under which 2-mode Gaussian density matrices become pure states. Possible pure states include the 2-mode squeezed pure states commonly mentioned in the literature, plus other pure states never mentioned before. We discuss the entropy and thermodynamic laws (Second Law, Fundamental Equation, and Gibbs-Duhem Equation) for the 2-mode states being considered. We study the change in entropy that is produced when a 2-mode Gaussian state is subjected to a measurement of the complex amplitude of one of its two modes. We derive upper and lower bounds for the final (i.e., after the measurement) entropy of the unmeasured mode, and we give necessary and sufficient conditions for the achievement of these bounds. The existence of the bounds is shown to be a consequence of the concavity property of the entropy function.

DOI
10.1142/s0217979292000839

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In this paper, we generalize to 2-mode states the 1-mode state results obtained in a previous paper. We study 2-mode Gaussian density matrices (i.e., density matrices of the form: exponential of a quadratic polynomial in the creation and annihilation operators for the two modes). We find a linear transformation which maps the two annihilation operators, one for each mode, into two new annihilation operators that are uncorrelated and unsqueezed. This allows us to express the density matrix as a product of two 1mode density matrices. We find general conditions under which 2-mode Gaussian density matrices become pure states. Possible pure states include the 2-mode squeezed pure states commonly mentioned in the literature, plus other pure states never mentioned before. We discuss the entropy and thermodynamic laws (Second Law, Fundamental Equation, and Gibbs-Duhem Equation) for the 2-mode states being considered. We study the change in entropy that is produced when a 2-mode Gaussian state is subjected to a measurement of the complex amplitude of one of its two modes. We derive upper and lower bounds for the final (i.e., after the measurement) entropy of the unmeasured mode, and we give necessary and sufficient conditions for the achievement of these bounds. The existence of the bounds is shown to be a consequence of the concavity property of the entropy function.
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Coordinate wave functions for the one-mode squeezed states produced by the quantum analog of the general linear transformation in phase space are calculated. The probability density [\ensuremath{\Vert}\ensuremath{\psi}(q)${\ensuremath{\Vert}}^{2}$] for these states is Gaussian with center predicted by the classical transformation. The quantum image (which includes the traditional two-mode squeeze operator) of a three-parameter symplectic map in two-mode phase space equally generates squeezed states having Gaussian \ensuremath{\Vert}\ensuremath{\psi}(q1{q}_{1}q1​,q1{q}_{1}q1​)${\ensuremath{\Vert}}^{2}$. The center of the two-mode Gaussian is again predicted by the classical mapping.
By the complex multimode Bogoliubov transformation, we obtain the general forms of squeeze operators and squeezed states including squeezed vacuum states, squeezed coherent states, squeezed Fock states and squeezed coherent Fock states, for a general multimode boson system. We decompose the squeezed operator into disentangling form in normal ordering to simplify the expressions of the squeezed states. We also calculate the statistical properties of the SS. Furthermore we prove that if it is non-degenerate, a Bogoliubov transformation matrix can be decomposed into three basic matrix, by which we can not only check the criterion of the minimum uncertainty state (MUS) found by Milburn, but also prove that except the special cases, any multimode squeezed state is MUS after the original creation and annihilation operators rotate properly. We also discuss some special cases of the three basic matrices. Finally we give an analytical results of a two-mode system as an example.