Margarita A. Man’ko

Probability representation and new entropic uncertainty relations for symplectic and optical tomograms

Entropic uncertainty relations for Shannon entropies associated with tomographic probability distributions of continuous quadratures are reviewed. New entropie uncertainty relations in the form of inequalities for integrals containing the tomograms of quantum states and deformation parameter are obtained.

Tomography of solitons

We develop the tomographic representation of wavefunctions which are solutions of the generalized nonlinear Schrodinger equation (NLSE) and show its connection with the Weyl–Wigner map. The generalized NLSE is presented in the form of a nonlinear Fokker–Planck-type equation for the standard probability distribution function (tomogram). In particular, this theory is applied to solitons, where tomograms for envelope bright solitons of a family of modified NLSEs are presented and numerically evaluated. Examples of symplectic tomography and Fresnel tomography of linear and nonlinear signals are discussed.

Nonstationary linear spin systems in the probability representation

The probability representation of angular momentum states and the connection of the representation with the formalism of the star-product quantization procedure are reviewed. The Schrodinger equation of a general time-dependent Hamiltonian, linear in angular momentum operators, and the evolution equation for the density operator in the probability representation are solved analytically by means of the formalism of linear time-dependent constants of motion. These analytical solutions define wavefunctions and tomograms of states (generic Dicke states), which contain the atomic coherent states as a particular case. The statistical properties of these new states are also evaluated. General forms of analytically solvable Hamiltonians are established in terms of the Euler angle parametrization of the three-dimensional rotations.

Evolution and Entanglement of Gaussian States in the Parametric Amplifier

We obtain the linear time-dependent constants of motion of the parametric amplifier and use them to determine the evolution of a general two-mode Gaussian state in the tomographic-probability representation. By means of the discretization of the continuous variable density matrix, we calculate the von Neumann and linear entropies to measure the entanglement properties between the modes of the amplifier. We compare the obtained results for the nonlocal correlations with those associated to a linear map of discretized symplectic Gaussian-state tomogram onto a qubit tomogram. We use this qubit portrait procedure to establish Bell-type inequalities, which provide a necessary condition to determine the separability of quantum states, which can be evaluated through homodyne detection. We define the other no-signaling nonlocal correlations through the portrait procedure for noncomposite systems.

Optical, symplectic and fresnel tomographies of quantum states

The description of photon quantum states by means of probability-distribution functions (tomograms) of three different kinds (optical, symplectic and Fresnel ones) is presented. Mutual relations between the optical, symplectic and Fresnel tomograms are established. Evolution equation for states of Bose-Einstein condensate (Gross-Pitaevskii nonlinear equation) is given in the tomographic-probability representation. Entropy of solitons related to the Shannon entropy of the tomographic-probability representation is considered.

Canonical transforms, quantumness and probability representation of quantum mechanics

The linear canonical transforms of position and momentum are used to construct the tomographic probability representation of quantum states where the fair probability distribution determines the quantum state instead of the wave function or density matrix. The example of Moshinsky shutter problem is considered.