Olga V. Man'ko

Alternative commutation relations, star products and tomography

Invertible maps from operators of quantum observables onto functions of c-number arguments and their associative products are first assessed. Different types of maps such as the Weyl-Wigner-Stratonovich map and s-ordered} quasi-distribution are discussed. The recently introduced symplectic tomography map of observables (tomograms) related to the Heisenberg-Weyl group is shown to belong to the standard framework of the maps from quantum observables onto the c-number functions. The star product for symbols of the quantum observable for each one of the maps (including the tomographic map) and explicit relations among different star products are obtained. Deformations of the Moyal star product and alternative commutation relations are also considered.

Star products, duality and double Lie algebras

Quantization of classical systems using the star-product of symbols of observables is discussed. In the star-product scheme an analysis of dual structures is performed and a physical interpretation is proposed. At the Lie algebra level duality is shown to be connected to double Lie algebras. The analysis is specified to quantum tomography. The classical tomographic Poisson bracket is found.

Tomographic representation of spin and quark states

We present a short review of the general principles of constructing tomograms of quantum states. We derive a general tomographic reconstruction formula for the quantum density operator of a system with a dynamical Lie group. In the reconstruction formula, the multiplicity of irreducible representation in Clebsch–Gordan decomposition is taken into account. Various approaches to spin tomography are discussed. An integral representation for the tomographic probability is found and a contraction of the spin tomogram to the photon-number tomography distribution is considered. The case of SU(3) tomography is discussed with the examples of quark states (related to the simplest triplet representations) and octet states.

SYMPLECTIC TOMOGRAPHY OF NONLINEAR COHERENT STATES OF A TRAPPED ION

Abstract The squeezed and rotated quadrature of an ion in a Paul trap is discussed in connection with reconstructing its quantum state using the symplectic tomography method. Marginal distributions of the quadrature for squeezed and correlated states and for nonlinear coherent states of a trapped ion are obtained and the density matrices in the Fock basis are expressed explicitly in terms of these marginal distributions.

Hilbert-Schmidt distance and non-classicality of states in quantum optics

Abstract The Hilbert—Schmidt distance between two arbitrary normalizable states is discussed as a measure of the similarity of the states. Unitary transformations of both states with the same unitary operator (e.g. the displacement of both states in the phase plane by the same displacement vector and squeezing of both states) do not change this distance. The nearest distance of a given state to the whole set of coherent states is proposed as a quantitative measure of non-classicality of the state which is identical when considering the coherent states as the most classical ones among pure states and the deviations from them as non-classicality. The connection to other definitions of the non-classicality of states is discussed. The notion of distance can also be used for the definition of a neighbourhood of considered states. Inequalities for the distance of states to Fock states are derived. For given neighbourhoods, they restrict common characteristics of the state as the dispersion of the number operator and the squared deviation of the mean values of the number operator for the considered state and the Fock state. Possible modifications in the definition of non-classicality for mixed states with dependence on the impurity parameter and by including the displaced thermal states as the most classical reference states are discussed.

The Pauli equation for probability distributions

The tomographic-probability distribution for a measurable coordinate and spin projection is introduced to describe quantum states as an alternative to the density matrix. An analogue of the Pauli equation for the spin-½ particle is obtained for such a probability distribution instead of the usual equation for the wavefunction. Examples of the tomographic description of Landau levels and coherent states of a charged particle moving in a constant magnetic field are presented.

Time-dependent oscillator with Kronig-Penney excitation

Abstract Exact solutions of the time-dependent Schrodinger equation for a quantum oscillator subject to periodical frequency δ-kicks are obtained. We show that the oscillator occurs in the squeezed state and calculate the corresponding squeezing coefficients and the energy increase rate in terms of Chebyshev polynomials.

Does the uncertainty relation determine the quantum state

The example of nonpositive trace-class Hermitian operator for which Robertson-Schrodinger uncertainty relation is fulfilled is presented. The partial scaling criterion of separability of multimode continuous variable system is discussed in the context of using nonpositive maps of density matrices.

Thermal noise and oscillations of the photon distribution for squeezed and correlated light

Abstract The oscillations of the photon distribution function for squeezed and correlated light are shown to decrease when the temperature increases. The influence of the squeezing parameter and photon quadrature correlation coefficient on the photon distribution oscillations at nonzero temperatures is studied. The connection of the deformation of the Planck distribution formula with oscillations of the photon distribution for squeezed and correlated light is discussed.The oscillations of photon distribution function for squeezed and correlated light are shown to decrease when the temperature increases.The influence of the squeezing parameter and photon quadrature correlation coefficient on the photon distribution oscillations at nonzero temperatures is studied. The connection of deformation of Planck distribution formula with oscillations of distribution for squeezed and correlated light is discussed.

TOMOGRAPHIC MAP WITHIN THE FRAMEWORK OF STAR-PRODUCT QUANTIZATION

Tomograms introduced for the description of quantum states in terms of probability distributions are shown to be related to a standard star-product quantization with appropriate kernels. Examples of symplectic tomograms and spin tomograms are presented.