V.I. Man'ko

f-oscillators and nonlinear coherent states

The notion of f-oscillators generalizing q-oscillators is introduced. For classical and quantum cases, an interpretation of the f-oscillator is provided as corresponding to a special nonlinearity of vibration for which the frequency of oscillation depends on the energy. The f-coherent states (nonlinear coherent states) generalizing q-coherent states are constructed. Applied to quantum optics, photon distribution function, photon number means, and dispersions are calculated for the f-coherent states as well as the Wigner function and Q-function. As an example, it is shown how this nonlinearity may affect the Planck distribution formula.

An introduction to the tomographic picture of quantum mechanics

Starting from the famous Pauli problem on the possibility of associating quantum states with probabilities, the formulation of quantum mechanics in which quantum states are described by fair probability distributions (tomograms, i.e. tomographic probabilities) is reviewed in a pedagogical style. The relation between the quantum state description and the classical state description is elucidated. The difference between those sets of tomograms is described by inequalities equivalent to a complete set of uncertainty relations for the quantum domain and to non-negativity of probability density on phase space in the classical domain. The intersection of such sets is studied. The mathematical mechanism that allows us to construct different kinds of tomographic probabilities like symplectic tomograms, spin tomograms, photon number tomograms, etc is clarified and a connection with abstract Hilbert space properties is established. The superposition rule and uncertainty relations in terms of probabilities as well as quantum basic equations like quantum evolution and energy spectra equations are given in an explicit form. A method to check experimentally the uncertainty relations is suggested using optical tomograms. Entanglement phenomena and the connection with semigroups acting on simplexes are studied in detail for spin states in the case of two-qubits. The star-product formalism is associated with the tomographic probability formulation of quantum mechanics.

Positive distribution description for spin states

Abstract We investigate a possibility of describing spin states in terms of a positive distribution function depending on continuous variables like Eulers angles. A spin state reconstruction procedure similar to the symplectic tomography is considered. A quantum evolution equation for the classical-like positive distribution function is found. Generalization to arbitrary values of angular momentum is discussed.

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.

Coherent states and excitation of N-dimensional non-stationary forced oscillator

Abstract New invariants, coherent states and transition amplitudes are constructed for non-stationary forced oscillators. The Franck-Condon factor is calculated.

Generation of squeezed states in a resonator with a moving wall

Abstract The problem of generation of squeezed states of the electromagnetic field in a one-dimensional resonator with a moving wall is considered. The squeezing and correlation coefficients for a harmonic law of motion of the wall are calculated.

Integrals of the motion, green functions, and coherent states of dynamical systems

The connection between the integrals of the motion of a quantum system and its Green function is established. The Green function is shown to be the eigenfunction of the integrals of the motion which describe initial points of the system trajectory in the phase space of average coordinates and moments. The explicit expressions for the Green functions of theN-dimensional system with the Hamiltonian which is the most general quadratic form of coordinates and momenta with time-dependent coefficients is obtained in coordinate, momentum, and coherent states representations. The Green functions of the nonstationary singular oscillator and of the stationary Schrödinger equation are also obtained.

Correlation functions of quantum q-oscillators

Abstract A nonlinearity of electromagnetic field vibrations described by q -oscillators is shown to produce an essential dependence of second order correlation functions on the intensity and deformation of the Planck distribution. Experimental tests of such a nonlinearity are suggested.

INVARIANTS AND THE EVOLUTION OF COHERENT STATES FOR A CHARGED PARTICLE IN A TIME-DEPENDENT MAGNETIC FIELD.

Abstract New invariants, coherent states and the Green function are constructed for a charged particle in a time-dependent magnetic field.

Classical and quantum Fisher information in the geometrical formulation of quantum mechanics

The tomographic picture of quantum mechanics has brought the description of quantum states closer to that of classical probability and statistics. On the other hand, the geometrical formulation of quantum mechanics introduces a metric tensor and a symplectic tensor (Hermitian tensor) on the space of pure states. By putting these two aspects together, we show that the Fisher information metric, both classical and quantum, can be described by means of the Hermitian tensor on the manifold of pure states.