Radon transform and pattern functions in quantum tomography

Abstract

The two-dimensional Radon transform of the Wigner quasiprobability is introduced in canonical form and the functions playing a role in its inversion are discussed. The transformation properties of this Radon transform with respect to displacement and squeezing of states are studied and it is shown that the last is equivalent to a symplectic transformation of the variables of the Radon transform with the contragredient matrix to the transformation of the variables in the Wigner quasiprobability. The reconstruction of the density operator from the Radon transform and the direct reconstruction of its Fock-state matrix elements and of its normally ordered moments are discussed. It is found that for finite-order moments the integration over the angle can be reduced to a finite sum over a discrete set of angles. The reconstruction of the Fock-state matrix elements from the normally ordered moments leads to a new representation of the pattern functions by convergent series over even or odd Hermite polynomials which is appropriate for practical calculations. The structure of the pattern functions as first derivatives of the products of normalizable and nonnormalizable eigenfunctions to the number operator is considered from the point of view of this new representation.

Molecular Tomography of the Quantum State by Time-Resolved Electron Diffraction

“…It is an advantage of the procedure that the requisite molecular parameters may be refined from the experimental data without any information on the potential energy surface. The probability density functions obtained in this way were then sampled with the functions ( ) [34]:…”

“…Approximate Procedures. When the TRED measurements do not span a complete cycle and the data are incomplete, only the diagonal density matrix elements, = , can be evaluated from the probability density ⟨ ( , )⟩ averaged over the time interval ≫ sup(| − | −1 ), with ̸ = , as shown by Richter and Wünsche [31][32][33][34], Leonhardt [29,35], Leonhardt and Raymer [36], and others [37][38][39]. This method uses the superposition principle combined with the structure of Schrödinger's equation to reveal the quantum state of a molecule from the time-averaged coordinatedependent probability density function.…”

Renormalized quantum tomography

“…The dual G(x, φ) provides also new pattern functions for the matrix elements, as previously noticed in Refs. [12,18]. For example, for the vacuum component one has…”

New properties of the V-line Radon transform and their imaging applications

“…Partial Differential Equations [2], probabilities [3], group theory [4], differential geometry [5]). In physics, it has opened a new investigation path to quantum states in phase space [6,7], to particle dynamics in accelerators [8]. In imaging technologies, it has covered medical imaging [9], astrophysics [10], non-destructive control [11], geophysical prospection [12], etc.…”