V. I. Man’ko

A tomographic setting for quasi-distribution functions

The method of constructing the tomographic probability distributions describing quantum states in parallel with density operators is presented. Known examples of Husimi-Kano quasi-distribution and photon number tomography are reconsidered in the new setting. New tomographic schemes based on coherent states and nonlinear coherent states of deformed oscillators, including q-oscillators, are suggested. The associated identity decompositions providing Gram-Schmidt operators are explicitly given.

Generalized tomographic maps

We introduce several possible generalizations of tomography to curved surfaces. We analyze different types of elliptic, hyperbolic, and hybrid tomograms. In all cases it is possible to consistently define the inverse tomographic map. We find two different ways of introducing tomographic sections. The first method operates by deformations of the standard Radon transform. The second method proceeds by shifting a given quadric pattern. The most general tomographic transformation can be defined in terms of marginals over surfaces generated by deformations of complete families of hyperplanes or quadrics. We discuss practical and conceptual perspectives and possible applications.

Optical tomography of photon-added coherent states, even and odd coherent states, and thermal states

Explicit expressions for optical tomograms of photon-added coherent states, even and odd photon-added coherent states, and photon-added thermal states are given in terms of Hermite polynomials. Suggestions for experimental homodyne detection of the considered photon states are presented.

Radon transform on the cylinder and tomography of a particle on the circle

The tomographic probability distribution on the phase space (cylinder) related to a circle or an interval is introduced. The explicit relations of the tomographic probability densities and the probability densities on the phase space for the particle motion on a torus are obtained, and the relation of the suggested map to the Radon transform on the plane is elucidated. The generalization to the case of a multidimensional torus is elaborated, and the geometrical meaning of the tomographic probability densities as marginal distributions on the helix discussed.

The geometry of density states, positive maps and tomograms

The positive and not completely positive maps of density matrices, which are contractive maps, are discussed as elements of a semigroup. A new kind of positive map (the purification map), which is nonlinear map, is introduced. The density matrices are considered as vectors, linear maps among matrices are represented by superoperators given in the form of higher dimensional matrices. Probability representation of spin states (spin tomography) is reviewed and U(N)-tomogram of spin states is presented. Properties of the tomograms as probability distribution functions are studied. Notion of tomographic purity of spin states is introduced. Entanglement and separability of density matrices are expressed in terms of properties of the tomographic joint probability distributions of random spin projections which depend also on unitary group parameters. A new positivity criterion for hermitian matrices is formulated. An entanglement criterion is given in terms of a function depending on unitary group paramete rs and semigroup of positive map parameters. The function is constructed as sum of moduli of U(N)- tomographic symbols of the hermitian matrix obtained after action on the density matrix of composite system by a positive but not completely positive map of the subsystem density matrix. Some two-qubit and two-qutritt states are considered as examples of entangled states. The connection with the star-product quantisation is discussed. The structure of the set of density matrices and their relation to unitary group and Lie algebra of the unitary group are studied. Nonlinear quantum evolution of state vector obtained by means of applying purification rule of density matrices evolving via dynamical maps is considered. Some connection of positive maps and entanglement with random matrices is discussed and used.

Parametric Excitation of Photon-added Coherent States

We study the evolution of the photon-added coherent state | α, m (introduced by Agarwal and Tara [Phys. Rev. A43, 492 (1991)]) due to a time dependence of the frequency of the electromagnetic field oscillator in a cavity or a vibrational frequency of an ion inside an electromagnetic trap. We give explicit expressions for the photon distribution function, mean values and variances of the quadrature components and of the photon number, the Wigner and Q-functions, etc. We show that the parametric excitation leads to strong oscillations of the photon (phonon) distribution function and changes the subpoissonian photon statistics to the superpoissonian one. Besides, it enables to achieve a larger squeezing coefficient than in the usual squeezed states.

From Equations of Motion to Canonical Commutation Relations: Classical and Quantum Systems

We consider equations of motion for classical and quantum systems. It is shown that they do not determine uniquely the canonical commutation relations, neither at the classical level, nor at the quantum level. By using some of the alternative commutation relations as deformed ones, we consider the description of deformed systems, classical and quantum. In particular, by using deformed oscillators we deal with photon statistics in nonlinear coherent states and possible influence of deformations on electrostatics.

Evolution Equation for a Joint Tomographic Probability Distribution of Spin-1 Particles

The nine-component positive vector optical tomographic probability portrait of quantum state of spin-1 particles containing full spatial and spin information about the state without redundancy is constructed. Also the suggested approach is expanded to symplectic tomography representation and to representations with quasidistributions like Wigner function, Husimi Q−function, and Glauber-Sudarshan P−function. The evolution equations for constructed vector optical and symplectic tomograms and vector quasidistributions for arbitrary Hamiltonian are found. The evolution equations are also obtained in special case of the quantum system of charged spin-1 particle in arbitrary electro-magnetic field, which are analogs of non-relativistic Proca equation in appropriate representations. The generalization of proposed approach to the cases of arbitrary spin is discussed. The possibility of formulation of quantum mechanics of the systems with spins in terms of joint probability distributions without the use of wave functions or density matrices is explicitly demonstrated.

Generalized tomographic maps and star-product formalism

We elaborate on the notion of generalized tomograms, both in the classical and quantum domains. We construct a scheme of star-products of thick tomographic symbols and obtain in explicit form the kernels of classical and quantum generalized tomograms. Some of the new tomograms may have interesting applications in quantum optical tomography.

Calculating means of quantum observables in the optical tomography representation

We introduce a dual map from the special class A of quantum observables â to a special class of generalized functions a(X, ϕ). The class A includes all symmetrized polynomials in the canonical variables