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Two and three coupled spins in magnetic field

Dmitry Gitman, Mario Cesar Baldiotti, Marcelo Meireles dos Santos

Published Apr 14, 2014• 0 citations
Publisher

Abstract

We consider three spin systems in the parallel magnetic fields and present some of possible exact solutions.In addition, we discuss an analogue of the Rabi problem for three-spin system.In order to do this, we reduce the problem of three coupled spins, an 8-level system, to a problem of a 3-level system in an effective field.We explicitly construct the evolution operator of the system in two special cases: when the fields and interaction are the same in all the spins but with a arbitrary time-dependence of the fields, and when the spins are subjected to a circular magnetic field (the Rabi problem).Some applications of these results are discussed.

DOI
10.22323/1.200.0009

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We consider three spin systems in the parallel magnetic fields and present some of possible exact solutions.In addition, we discuss an analogue of the Rabi problem for three-spin system.In order to do this, we reduce the problem of three coupled spins, an 8-level system, to a problem of a 3-level system in an effective field.We explicitly construct the evolution operator of the system in two special cases: when the fields and interaction are the same in all the spins but with a arbitrary time-dependence of the fields, and when the spins are subjected to a circular magnetic field (the Rabi problem).Some applications of these results are discussed.
We study the four-level-system given by two quantum dots immersed in a time-dependent magnetic field, which are coupled to each other by an effective Heisenberg-type interaction. We describe the construction of the corresponding evolution operator in a special case of different time-dependent parallel external magnetic fields. We find a relation between the external field and the effective interaction function. The obtained results are used to analyze the theoretical implementation of a universal quantum gate.
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We propose a method to describe the evolution of two spins coupled by hyperfine i nteraction in an external time- dependent magnetic field. We apply the approach to the case of hyperfine interaction with axial symmetry, which can be solved exactly in a constant, appropriately oriented magnetic field. In order to t reat t he n onstationary d ynamical p roblem, we modify the time-dependent Schrödinger equation through a change of representation that, by exploiting an instantaneous (adiabatic) basis makes the time-dependent Hamiltonian diagonal at any time instant. The solution of the transformed time-dependent Schrödinger FRVBUJPO in the form of chronologically ordered exponents with transparent pre-exponential coefficients is reported. This solution is highly simplified when an adiabatically varying magnetic field perturbs the system. The approach here proposed may be used for the perturbative treatment of other dynamical problems with no exact solution.
The evolution operator of the spin–orbit coupled system in an external magnetic field is factorized according to the operator equation:[Formula: see text]where HSO is the spin–orbit coupled Hamiltonian and HZ is the Zeeman Hamiltonian. The preceding equation is expressed in matrix form using the theory of the O(4) group representations. For the special case of [Formula: see text], an explicit form of the matrix C(t) is found. The operator corresponding to the matrix C(t) is found in a closed form using the projection operators constructed in the Appendix. Two special cases, corresponding to a weak and a strong external field, are considered, and possible applications of the results obtained are indicated.