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Journal of Optics B Quantum and Semiclassical Optics

Tomography of photon-added and photon-subtracted states

M. R. Bazrafkan, Vladimir I. Man’ko

Published Jul 11, 2003• 7 citations
Publisher

Abstract

The purpose of this paper is to introduce symplectic and optical tomograms of photon-added and photon-subtracted quantum states. Explicit relations for the tomograms of photon-added and photon-subtracted squeezed coherent states and squeezed number states are obtained. Generating functions for the moments of the tomographic probability functions obtained are studied.

DOI
10.1088/1464-4266/5/4/306

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The purpose of this paper is to introduce symplectic and optical tomograms of photon-added and photon-subtracted quantum states. Explicit relations for the tomograms of photon-added and photon-subtracted squeezed coherent states and squeezed number states are obtained. Generating functions for the moments of the tomographic probability functions obtained are studied.
A review of the symplectic, optical tomography and photon number tomography is presented. Both the set of quantum states and the set of classical states described by symplectic tomographic probability distributions (tomograms) are studied, and it is shown that the sets have common part but there exist tomograms of classical states which are not admissible in quantum mechanics and, vica versa, there exist tomograms of quantum states which are not admissible in classical mechanics. Negative probabilities in classical and quantum mechanics corresponding to tomograms of classical and quantum states are compared with properties of nonpositive and nonnegative density operators, respectively.
A review of the symplectic, optical tomography and photon number tomography is presented. Both the set of quantum states and the set of classical states described by symplectic tomographic probability distributions (tomograms) are studied, and it is shown that the sets have common part but there exist tomograms of classical states which are not admissible in quantum mechanics and, vica versa, there exist tomograms of quantum states which are not admissible in classical mechanics. Negative probabilities in classical and quantum mechanics corresponding to tomograms of classical and quantum states are compared with properties of nonpositive and nonnegative density operators, respectively.
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.
Optical tomogram of a quantum state can serve as an alternative to its density matrix since it contains all the information on this state. In this paper we use the explicit expression of the intermediate state | q, f , g〉(that is, an eigenvector of the coordinate and momentum operator fQ + gP ) and the Radon transform between the Wigner operator Δ(p, q) and the pure-state density operator | q, f, g 〉〈 q, f, g | to calculate the specific tomograms for multiple-photon-added coherent states (MPACSs), thermal states (MPATSs) and displaced thermal states (MPADTSs). Their analytical tomograms are actually the finite-summation representations related to single-variable Hermite polynomials. Besides, the numerical results indicate that the larger photon-addition number can lead to the multi-peak tomographic distributions for these three states, and the measurable tomograms of MPACSs and MPADTSs exhibit π-periodic features but MPATSs can’t.