Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Canadian Journal of Physics

Factorization of the evolution operator for a spin–orbit coupled system in an external magnetic field for an arbitrary L and

R. Bogdanović, M. A. Whitehead, M. S. Gopinathan

Published Dec 1, 1976• 0 citations
Publisher

Abstract

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.

DOI
10.1139/p76-275

Related papers

One source of spin–orbit coupling in the Schrödinger representation

R. Chatterjee, J.M. Dixon

The origin of spin–orbit coupling has been investigated in the Schrödinger picture with the constraint of Galilean invariance. A Lagrangian has been constructed with this group symmetry which contains a term that has been interpreted as the damping of the translational motion of the electron in a spinor field. From the Lagrangian a Hamiltonian [Formula: see text] has been formed by classical methods. A unitary transformation has been found which transforms [Formula: see text] into a Hamiltonian [Formula: see text] which contains not only the Zeeman terms, with the correct magnetic moment, but also the spin–orbit coupling.

Canadian Journal of Physics•Dec 1, 1979•0 citations

Application of Product Operator Formalism to the Strongly Coupled Spin (I=1/2) Systems

Miao Xi, Jie Hu

Product operator formalism has been developed to evaluate, in closed analytical form, the time evolution for strongly coupled spin (I=1/2) systems. This formalismis based on two facts: (ⅰ) the Hamiltonian for a strongly coupled spin system is a zero-quantum operator and when an arbitrary zero-quantum operator acts on a p-quantum operator, what is yielded is still a p-quantum operator and can be expressed in terms of a linear combination of a complete p-quantum operator base set of the spin system; (ⅱ) the zeroquantum and unitary transformation leads the Hamiltonian to be only a linear combination of zero-quantum longitudinal magnetization and spin order operators. Thus the time evolution for the spin systems can be evaluated in closed analytical form. The formalism retains completely the original character of the product operator formalism and enlarges its applicability. It can deal with both strongly and weakly coupled spin (I=1/2) systems in united and closed analytical forms.

Science China Mathematics•Jan 1, 1993•0 citations

Two and three coupled spins in magnetic field

Dmitry Gitman, Mario Cesar Baldiotti, Marcelo Meireles dos Santos

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.

•Apr 14, 2014•0 citations

Two interacting spins in external fields and application to quantum computation

Mario Cesar Baldiotti, Vladislav G. Bagrov, Dmitri M Gitman

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.

Physics of Particles and Nuclei Letters•Nov 19, 2009•5 citations

A new method of calculating the unitary evolution matrix ds(t) of the spin-s operators and its applications

Ming-Liang Hu, Xiao-Qiang Xi, 西安邮电学院应用数理系，西安 710061

We propose a method for calculating the unitary evolution matrix of the arbitrary spin-s operators rigorously. This is an indirect method, which differs from the method of group theory or the method of direct calculation. The kernel of our method is to use the identity of two systems in their expressions, namely the Hamiltonian Hs=Sx of spin-s particle and the Hamiltonian of the Heisenberg XX open chain with interaction J=n(N-n). Because of this identity, the calculation of the unitary evolution matrix of the spin-s operators is substituted by the calculation of the state evolution matrix in Heisenberg XX open chain. As examples, the unitary evolution matrix of s=3/2, s=2 and s=5/2 are calculated by using our method. Since the evolution of the state |sm〉 under the operator e-itSx corresponds to the rotation of the initial state |sm〉 around the x-axis by an angle β＝t, and the evolution matrix element dsm′m(t)=〈sm′|e-itSx|sm〉 is just the projection of the finial state e-itSx|sm〉 on the initial state |sm′〉, the evolution matrix at t=π corresponds the perfect transmission of quantum state in the Heisenberg XX open chain.

Acta Physica Sinica•Jan 1, 2008•3 citations

On the factorization method for the quantum statistical description of dynamics of an isolated spin system

Alexander Alexandrovitch Samokhin, А. В. Зыль, Николай Леонидович Замарашкин

We study the applicability of the formula that factors the trace of the diagonal part of spin operator products in the case of a relatively small number of particles of an isolated spin system. The validity of this formula for a large number of particles follows from the basic principles of quantum statistical mechanics. The spin system under consideration includes dipole–dipole interaction and the Zeeman interaction with an external magnetic field. We establish that the accuracy of this formula monotonically increases as the magnetic field increases. At the same time, the dependence on the number of particles in the range $$2\div10$$ for various configurations turns out to be sharply nonmonotone.

Theoretical and Mathematical Physics•Mar 1, 2024•0 citations
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.
The origin of spin–orbit coupling has been investigated in the Schrödinger picture with the constraint of Galilean invariance. A Lagrangian has been constructed with this group symmetry which contains a term that has been interpreted as the damping of the translational motion of the electron in a spinor field. From the Lagrangian a Hamiltonian [Formula: see text] has been formed by classical methods. A unitary transformation has been found which transforms [Formula: see text] into a Hamiltonian [Formula: see text] which contains not only the Zeeman terms, with the correct magnetic moment, but also the spin–orbit coupling.
Product operator formalism has been developed to evaluate, in closed analytical form, the time evolution for strongly coupled spin (I=1/2) systems. This formalismis based on two facts: (ⅰ) the Hamiltonian for a strongly coupled spin system is a zero-quantum operator and when an arbitrary zero-quantum operator acts on a p-quantum operator, what is yielded is still a p-quantum operator and can be expressed in terms of a linear combination of a complete p-quantum operator base set of the spin system; (ⅱ) the zeroquantum and unitary transformation leads the Hamiltonian to be only a linear combination of zero-quantum longitudinal magnetization and spin order operators. Thus the time evolution for the spin systems can be evaluated in closed analytical form. The formalism retains completely the original character of the product operator formalism and enlarges its applicability. It can deal with both strongly and weakly coupled spin (I=1/2) systems in united and closed analytical forms.
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.
We propose a method for calculating the unitary evolution matrix of the arbitrary spin-s operators rigorously. This is an indirect method, which differs from the method of group theory or the method of direct calculation. The kernel of our method is to use the identity of two systems in their expressions, namely the Hamiltonian Hs=Sx of spin-s particle and the Hamiltonian of the Heisenberg XX open chain with interaction J=n(N-n). Because of this identity, the calculation of the unitary evolution matrix of the spin-s operators is substituted by the calculation of the state evolution matrix in Heisenberg XX open chain. As examples, the unitary evolution matrix of s=3/2, s=2 and s=5/2 are calculated by using our method. Since the evolution of the state |sm〉 under the operator e-itSx corresponds to the rotation of the initial state |sm〉 around the x-axis by an angle β＝t, and the evolution matrix element dsm′m(t)=〈sm′|e-itSx|sm〉 is just the projection of the finial state e-itSx|sm〉 on the initial state |sm′〉, the evolution matrix at t=π corresponds the perfect transmission of quantum state in the Heisenberg XX open chain.
We study the applicability of the formula that factors the trace of the diagonal part of spin operator products in the case of a relatively small number of particles of an isolated spin system. The validity of this formula for a large number of particles follows from the basic principles of quantum statistical mechanics. The spin system under consideration includes dipole–dipole interaction and the Zeeman interaction with an external magnetic field. We establish that the accuracy of this formula monotonically increases as the magnetic field increases. At the same time, the dependence on the number of particles in the range 2÷102\div102÷10 for various configurations turns out to be sharply nonmonotone.