Center-of-mass tomography and probability representation of quantum states for tunneling

Abstract

We develop a representation of quantum states in which the states are described by fair probability distribution functions instead of wave functions and density operators. We present a one-random-variable tomography map of density operators onto the probability distributions, the random variable being analogous to the center-of-mass coordinate considered in reference frames rotated and scaled in the phase space. We derive the evolution equation for the quantum state probability distribution and analyze the properties of the map. To illustrate the advantages of the new tomography representations, we describe a new method for simulating nonstationary quantum processes based on the tomography representation. The problem of the nonstationary tunneling of a wave packet of a composite particle, an exciton, is considered in detail.

Quaternion Representation and Symplectic Spin Tomography

“…Starting from pioneer work in the field [4], parameterization through the Euler angles was suggested as a physically natural alternative to (9). The use of the Euler angles is one of the ways to represent the rotation of R 3 .…”

A use of the central limit theorem to obtain a classical limit for the center-of-mass tomogram

“…, X N ) T , H n denotes the Hermite polynomial and we put m = Ω = 1 for the mass as well as the frequency. In the case, taking into account that we do not put = 1 like it was done in [2,3]) we get…”

“…In [1] we studied the center-of-mass tomogram [2,3] in the limiting case of many degrees of freedom N → +∞. It was shown that the final distribution tends to Gaussian if the state of the initial system is a product of excited states of a quantum oscillator and some additional conditions are satisfied.…”

“…, (q N , pN ). Following to [2,3] we define the center-of-mass tomogram ω cm of a quantum state ρ ∈ S(H) as a function of 2N + 1 parameters µ = (µ1, . .…”

Tomograms for open quantum systems: In(finite) dimensional optical and spin systems

“…In another line of studies, simulation of quantum systems have been performed using tomography. For example, tomograms were used for simulation of tunneling [25][26][27] and multimode quantum states [28]. Attempts have also been made to understand the tomogram via path integrals [29,30].…”