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Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE

Application of quaternion algebra for calculation of photon echo signal

Anatolii N. Leukhin, Ivan I. Popov

Published Apr 20, 2004• 1 citation
Publisher

Abstract

Mathematical formalism of photon echo exciting at interaction of two-level quantum system with an electromagnetic field within the framework of quaternion algebra is offered. The rotation quaternion is used instead of density operator for representation of interaction dynamics of an exterior field with two-level atom, where the components of rotation quaternion are the coordinates of Bloch vector. Liouville fon-Neumann equation for density matrix evolution is substituted by the differential quaternion equation in a quaternion model. The expression for rotation quaternion appropriating to an unitary operator of rotation in matrix representation is obtained. It is shown that the solution of the differential quaternion equation does not require the difficult mathematical operation connected to an evaluation of an exponent from a matrix. The expression for a quaternion superoperator is obtained.

DOI
10.1117/12.562205

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Mathematical formalism of photon echo exciting at interaction of two-level quantum system with an electromagnetic field within the framework of quaternion algebra is offered. The rotation quaternion is used instead of density operator for representation of interaction dynamics of an exterior field with two-level atom, where the components of rotation quaternion are the coordinates of Bloch vector. Liouville fon-Neumann equation for density matrix evolution is substituted by the differential quaternion equation in a quaternion model. The expression for rotation quaternion appropriating to an unitary operator of rotation in matrix representation is obtained. It is shown that the solution of the differential quaternion equation does not require the difficult mathematical operation connected to an evaluation of an exponent from a matrix. The expression for a quaternion superoperator is obtained.
Abstract The solution to the time-dependent Pauli–Schrödinger equation is written using quaternion algebra, i.e. by expressing the unitary operator describing the time evolution of the Pauli spinor and the Pauli spinor itself as left-chirality quaternions. The quaternion approach is used to express the time evolution of a qubit on the Bloch sphere under the action of the commonly used quantum gates and to implement the two-qubit controlled-NOT (CNOT) gate and the three-qubit Tofolli gate. The quaternion formalism is also applied to describe: (1) a general two-qubit quaternionic gate, (2) the Larmor precession of an electron spin in an uniform magnetic field, and (3) the issue of measurements in quantum mechanics. Finally, the time-dependent Pauli–Schrödinger equation is also solved using right-chirality quaternions.
This work is dedicated to investigation the properties and opportunities of photon echo with the purpose of development the physical principle of embodying the optical evaluations on a superposition state of the group of excited particles of resonant gas, which could serve as a basis for increasing of features of quantum information processing regime which is supposed in work. One of the possible physical representations of a quaternion in the signal of photon echo is offered. The principle of the optical echo-processor operation is described to fulfill the operations with quantities which are introduced by quaternions. The example of optical transformation of quaternion quantity in the optical echo-processor is given.
An approximate numerical method is proposed and discussed for solving the evolution of the spin density operator when the quantum system has an interaction with an external electromagnetic field.In this method by separating the relaxation and field interaction processes at small steps, instead of solving the conventional Liouville-von Neumann or Bloch differential equations, the time evolution of the density operator is efficiently obtained by a two-stage numerical algorithm.Here we have compared the results of this approach with Bloch equation results for a twolevel quantum system.The proposed approach has potential applications in calculation of the time evolution for different atomic system including nuclear or electron spin resonance systems.