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

Features of application of quaternions at the solution of the Bloch equation

Anatolii N. Leukhin, Ivan I. Popov

Published Apr 20, 2004• 0 citations
Publisher

Abstract

Quaternion algebra is offered as the mathematical formalism of dynamics of interaction of a two-level quantum system with an electromagnetic field. Condition of two-level atom, densities operator, Bloch vector, “gyrating moment” vector are considered as the elements of a one-dimensional vector space of quaternion H1, i.e. are represented as quaternion. Bloch equations in a quaternion formalism are noted. Solution method of the differential quaternion Bloch equation is obtained.

DOI
10.1117/12.562203

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Quaternion algebra is offered as the mathematical formalism of dynamics of interaction of a two-level quantum system with an electromagnetic field. Condition of two-level atom, densities operator, Bloch vector, “gyrating moment” vector are considered as the elements of a one-dimensional vector space of quaternion H1, i.e. are represented as quaternion. Bloch equations in a quaternion formalism are noted. Solution method of the differential quaternion Bloch equation is obtained.
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.
The arguments of Chapter 2, as well as of those of subsequent chapters, have as their foundation the formalism based on the state vector of a system of which the evolution is described by the Schrödinger equation. In fact, such an approach is badly suited to the case in which the coupling between an atom and its environment (for example through collisions with other atoms or spontaneous emission into formerly empty modes of the electromagnetic field) cannot be neglected. If the correlations induced by these interactions between the atom and its environment do not concern us and we are only interested in the evolution of the atom, the formalism of the density matrix must be employed. This provides a description at all times of the state of the atom, although a state vector for the atom alone cannot be defined. In this formalism the effect of the environment on the atom is accounted for by the introduction of suitable relaxation terms (Section 2C.1) in the equation of evolution of the density matrix. An important application of the density matrix is to the case of a two-level atomic system for which the relaxation terms lead to its deexcitation to a level of lower energy. We shall show that in this case the density matrix can be represented by a vector, known as the Bloch vector , which will allow us to give simple geometrical pictures of the evolution of the system.
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.
The quantum theoretical description of microscopic systems is based on differential operators and the solution of the underlying eigenvalue equations. This approach is called Schrödinger approach or wave mechanics. Heisenberg developed another equivalent approach called matrix mechanics. In matrix mechanics all observables are represented by matrices. All possible eigenvalues of a matrix representing an observable are the possible outcomes of experimental measurement on that observable. This chapter gives the preliminary concepts relative to the use of the matrix method. It presents the various elements of matrix mechanics and discusses the matrix representation of operators of certain systems. The chapter introduces Dirac's notation of state vectors, Hilbert space, projection operator, displacement operators and the generators of translations. The theory of quaternions has found applications in classical mechanics, quantum mechanics and relativity. It has been used in aerospace applications, flight simulators, computer graphics, animation, visualization, fractals and virtual reality.
Abstract In the present article, we consistently develop the main issues of the Bloch vectors formalism for an arbitrary finite-dimensional quantum system. In the frame of this formalism, qudit states and their evolution in time, qudit observables and their expectations, entanglement and nonlocality, etc are expressed in terms of the Bloch vectors—the vectors in the Euclidean space R d 2 − 1 , arising under decompositions of observables and states in different operator bases. Within this formalism, we specify for all d ⩾ 2 the set of Bloch vectors of traceless qudit observables and describe its properties; also, find for the sets of the Bloch vectors of qudit states, pure and mixed, the new compact expressions in terms of the operator norms that explicitly reveal the general properties of these sets and have the unified form for all d ⩾ 2. For the sets of the Bloch vectors of qudit states under the generalized Gell‐Mann representation, these general properties cannot be analytically extracted from the known equivalent specifications of these sets via the system of algebraic equations. We derive the general equations describing the time evolution of the Bloch vector of a qudit state if a qudit system is isolated and if it is open and find for both cases the main properties of the Bloch vector evolution in time. For a pure bipartite state of a dimension d 1 × d 2 , we quantify its entanglement via the characteristics of the Bloch vectors for its reduced states. The introduced general formalism is important both for the theoretical analysis of quantum system properties and for quantum applications, in particular, for optimal quantum control, since, for systems where states are described by vectors in the Euclidean space, the methods of optimal control, analytical and numerical, are well developed.
By using the recent mathematical tools developed in quaternionic differential operator theory, we solve the Schrödinger equation in the presence of a quaternionic step potential. The analytic solution for the stationary states allows one to explicitly show the qualitative and quantitative differences between this quaternionic quantum dynamical system and its complex counterpart. A brief discussion on reflected and transmitted times, performed by using the stationary phase method, and its implication on the experimental evidence for deviations of standard quantum mechanics is also presented. The analytic solution given in this paper represents a fundamental mathematical tool to find an analytic approximation to the quaternionic barrier problem (up to now solved by numerical method).