Vladimir I. Man'ko

Symplectic tomography as classical approach to quantum systems

Abstract By using a generalization of the optical tomography technique we describe the dynamics of a quantum system in terms of equations for a purely classical probability distribution which contains complete information about the system.

Classical-like description of quantum dynamics by means of symplectic tomography

The dynamical equations of quantum mechanics are rewritten in the form of dynamical equations for the measurable, positive marginal distribution of the shifted, rotated, and squeezed quadrature introduced in the so-called “symplectic tomography”. Then the possibility of a purely classical description of a quantum system as well as a reinterpretation of the quantum measurement theory is discussed and a comparison with the well-known quasi-probabilities approach is given. Furthermore, an analysis of the properties of this marginal distribution, which contains all the quantum information, is performed in the framework of classical probability theory. Finally, examples of the harmonic oscillators states dynamics are treated.

Reconstructing the density operator by using generalized field quadratures

The Wigner function for one- and two-mode quantum systems is explicitly expressed in terms of the marginal distribution for the generic linearly transformed quadratures. Then, the density operator of those systems is also written in terms of the marginal distribution of these quadratures. Some applications and a reduction to the usual optical homodyne tomography are considered.

The quantum strong subadditivity condition for systems without subsystems

The strong subadditivity condition for the density matrix of a quantum system, which does not contain subsystems, is derived using the qudit-portrait method. An example of the qudit state in the seven-dimensional Hilbert space corresponding to spin j = 3 is presented in detail. New entropic inequalities in the form of the subadditivity condition and strong subadditivity condition for spin tomograms determining the qudit states are obtained and given on the example of j = 2 and 3.

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.

Density matrix from photon number tomography

We provide a simple analytic relation which connects the density operator of the radiation field with the number probabilities. The problem of experimentally sampling a general matrix element is studied, and the deleterious effects of nonunit quantum efficiency in the detection process are analyzed showing how they can be reduced by using the squeezing technique. The obtained result is particularly useful for intracavity field reconstruction states.

Non-classical properties of states generated by the excitations of even/odd coherent states of light

Photon-added even and odd coherent states of light are introduced. The mathematical and physical properties of such states are studied. The behaviour of the squeezing coefficient and the Mandel parameter is investigated. The quasiprobability distributions and the distribution of the field quadrature are studied in detail. The cumulants and factorial moments of the photon distribution are calculated. Strong oscillations of the ratio of the cumulant to factorial moment are demonstrated.

On the coherent states, displacement operators and quasidistributions associated with deformed quantum oscillators

The Wigner (W), Husimi-Kano (Q) and Glauber-Sudarshan (P) quasidistributions are generalized to f-deformed quasidistributions which extend the parametric family of s-ordered quasidistributions of Cahill and Glauber. The deformation procedure is obtained via a canonical nonisometric transform of the displacement operators which preserves the form of the standard creation-annihilation commutation relation, hence the Heisenberg-Weyl algebra, but changes the scalar product in the Hilbert space of the oscillator states. A whole class of new resolutions of the identity is introduced. The time evolution equation for the new generalized quasidistributions is derived.

Wigner's Problem and Alternative Commutation Relations for Quantum Mechanics

It is shown that for quantum systems the vector field associated with the equations of motion may admit alternative Hamiltonian descriptions, both in the Schrodinger and Heisenberg picture. We illustrate these ambiguities in terms of simple examples.

Phase space eigenfunctions of multidimensional quadratic hamiltonians

We obtain the explicit expressions for phase space eigenfunctions (PSE), i.e. Weyls symbols of dyadic operators like |n > , |m > being the solution of the Schrodinger equation with the Hamiltonian which is a quite arbitrary multidimensional quadratic form of the operators of Cartesian coordinates and conjugated to them momenta with time-dependent coefficients. It is shown that for an arbitrary quadratic Hamiltonian one can always construct the set of completely factorized PSE which are products of N factors, each factor being dependent only on two arguments for n≠m and on a single argument for n=m. These arguments are nothing but constants of motion of the correspondent classical system. PSE are expressed in terms of the associated Laguerre polynomials in the case of a discrete spectrum and in terms of the Airy functions in the continuous spectrum case. Three examples are considered: a harmonic oscillator with a time-dependent frequency, a charged particle in a nonstationary uniform magnetic field, and a particle in a time-dependent uniform potential field.