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Old and New Concepts of Physics

Bogoliubov Transform and Tomography of Nonclassical States of Trapped Ion

Olga V. Man’ko, Vladimir I. Man’ko

Published Jul 3, 2007• 0 citations
Publisher

Abstract

Bogoliubov transforms are discussed in some problems of quantum optics. Squeezed and rotated quadrature of an ion in a Paul trap is discussed in connection with reconstructing its quantum state using symplectic-tomography method. Symplectic and photon-number tomograms of the quadrature for squeezed and correlated states, for nonlinear coherent states and for odd/even coherent states of a trapped ion which are tomographic symbols of density operators of the states are reviewed.

DOI
10.2478/v10005-007-0019-6

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Bogoliubov transforms are discussed in some problems of quantum optics. Squeezed and rotated quadrature of an ion in a Paul trap is discussed in connection with reconstructing its quantum state using symplectic-tomography method. Symplectic and photon-number tomograms of the quadrature for squeezed and correlated states, for nonlinear coherent states and for odd/even coherent states of a trapped ion which are tomographic symbols of density operators of the states are reviewed.
A new type of tomogram, squeeze tomograms, of quantum states of light, is introduced, which is based on measuring photon statistics after the action of a squeeze operator on the states' density matrix. The introduced tomograms are photon number probability functions depending on two real parameters which are a scaling factor and a rotation angle. The rotation angle and scaling factor label the reference frame in the phase space of field quadratures in which the photon statistics is measured. The expressions of the squeeze tomogram in terms of the Wigner function and density matrix in quadrature representation are obtained. The examples of squeezed states, even and odd coherent states and thermal states are considered in detail. The multimode generalization is discussed.
States with explicit quantum character, such as squeezed vacuum and bright squeezed light, as well as coherent states and incoherent superpositions of coherent states were generated and analysed by tomographical methods. Wigner functions, photon-number distributions, density matrices and phase distributions were reconstructed with high accuracy. Features such as photon number oscillations, sub-Poissonian and super-Poissonian photon statistics, bifurcations of the phase distribution, and loss of coherence were observed, demonstrating the usefulness of quantum state reconstruction as an analysing tool in quantum optics experiments.
We show that the nonlinear quadrature-dependent Bogoliubov transformations proposed by Siena et al. [Phys. Rev. A 64, 063803 (2001)] can in fact be constructed by the combination of two unitary transformations, a quadrature-dependent displacement followed by the standard squeezed transformation. Such a decomposition turns a nonlinear problem into an essentially linear one so that we are able to express explicitly the mean values and deviations of the quadrature operators and the photon variables under the multiphoton states in terms of those quantities averaged over the standard squeezed states involving only the quadrature-independent Bogoliubov transformation.