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Wave function of the spin cluster in the external magnetic field obtained by group theory method

Xiao-Bo Yan

Published Jan 1, 2011• 0 citations
OpenAlex

Abstract

We have studied an anisotropic spin cluster of 3 spin 1/2 particles with antiferromagnetic exchange interaction with non-uniform coupling constants.This system has an algebraic structure.Using group theory method,we have obtained the rigorous analytical solution to the time dependent Schrdinger equation.

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We study an anisotropic spin cluster of 3 spin S = 1/2 particles with antiferromagnetic exchange interaction with non-uniform coupling constants. A time-dependent magnetic field is applied to control the time evolution of the cluster. It is well known that for an odd number of sites a spin cluster qubit can be defined in terms of the ground state doublet. The universal one-qubit logic gate can be constructed from the time evolution operator of the non-autonomous many-body system, and the six basic one-qubit gates can be realized by adjusting the applied time-dependent magnetic field.
The Heisenberg spin cluster with two particles controlled by a time-dependent magnetic field is investigated. Since the system possesses an SU（2） algebraic structure, the exact analytical solution to time evolution operator can be obtained by using algebraic dynamical method. Based on the analytical solution, the time evolution of wave function of the two-particle system can be obtained, then the entanglement can be calculated easily. The time evolution of the entanglement for different initial states is studied, and the effect of external field on the entanglement is discussed as well.
A homonuclear linear [Formula: see text] coupling spin cluster with the middle particle driven by an external time-dependent magnetic field is investigated by using the method of algebraic dynamics. The exact analytical solutions of the time-dependent Schrodinger equation of the spin cluster system are derived and employed to study the geometric phase. An alternative expression of the geometric phase in each eigenstate is obtained. It is shown that the geometric phase is related to the external magnetic-field parameter θ (the angle between the magnetic field and the Z axis) and the effective coupling strength Jn. Based on the relation, how the geometric phase depends on the coupling strength Jn in different reducible subspace is discussed.PACS Nos.: 33.20.Wr, 03.65.Fd, 03.65.Vf
The St¨uckelberg equation for a particle with two spin states, S = 1 and S = 0, is studied in the presence of an external uniform magnetic field. In relativistic case, the particle is described by an 11-component wave function. On the solutions of the equation, the operators of energy, the third projection of the total angular momentum, and the third projection of the linear momentum along the direction of the magnetic field are diagonalized. After separation of variables, we derive a system for 11 functions depending on one variable. We perform the nonrelativistic approximation in this system. For this we apply the known method of deriving nonrelativistic equations from relativistic ones, which is based on projective operators related to the matrix Γ0 of the relativistic equation. The nonrelativistic wave function turns out to be 4-dimensional. We derive the system for 4 functions. It is solved in terms of confluent hypergeometric functions. There arise three series of energy levels with corresponding solutions. This result agrees with that obtained for the relativistic St¨uckelberg equation.
We investigate the reversal of magnetization and the coherence of tunneling when an external magnetic field is rotated instantaneously in systems of a few (N) spin 1/2 particles described by an anisotropic Heisenberg Hamiltonian at T=0. The temporal evolution is calculated by a numerically exact solution of the time-dependent Schrödinger equation, and the mean value in time of each spin component is computed as a function of the magnetic field. The correlation function and the spectrum are analyzed in terms of the macroscopic quantum coherence. Our calculations demonstrate that this model for small (N<11) magnetic particles exhibit collective tunneling of the magnetization only for some specific resonant values of the applied magnetic field, at variance with the Stoner–Wohlfarth model.
We study the low energy states of finite spin chains with isotropic (Heisenberg) and anisotropic (XY and Ising-like) antiferromagnetic exchange interaction with uniform and nonuniform coupling constants. We show that for an odd number of sites a spin cluster qubit can be defined in terms of the ground state doublet. This qubit is remarkably insensitive to the placement and coupling anisotropy of spins within the cluster. One- and two-qubit quantum gates can be generated by magnetic fields and intercluster exchange, and leakage during quantum gate operation is small. Spin cluster qubits inherit the long decoherence times and short gate operation times of single spins. Control of single spins is hence not necessary for the realization of universal quantum gates.