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Acta Physica Sinica

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

Published Jan 1, 2008• 3 citations

Abstract

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.

DOI: 10.7498/aps.57.3319

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Abstract The not necessarily unitary evolution operator of a finite dimensional quantum system is studied with the help of a projection operators technique. Applying this approach to the Schrödinger equation allows the derivation of an alternative expression for the evolution operator, which differs from the traditional chronological exponent. An appropriate choice of projection operators results in the possibility of studying the diagonal and non-diagonal elements of the evolution operator separately. The suggested expression implies a particular form of perturbation expansion, which leads to a new formula for the short time dynamics. The new kind of perturbation expansion can be used to improve the accuracy of the usual chronological exponent significantly. The evolution operator for any arbitrary time can be efficiently recovered using the semigroup properties. The method is illustrated by two examples, namely the dynamics of a three-level system in two nonresonant laser fields and the calculation of the partition function of a finite XY-spin chain.

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.

A fully spin-adapted approach to many-electron density matrices is developed in the context of the unitary group approach to many-electron systems. An explicit expression for the single-electron spin-density operator, as a polynomial of degree two in the orbital U(n) generators, is derived for the case of spin-independent systems. Extensions to spin-dependent systems are also considered, leading to the appearance of total-spin transition densities, whose general properties are investigated. A corresponding formalism for the two-electron density matrix, which is capable of further generalization, is also developed. The results of this paper, together with recent developments on the matrix elements of the U(2n) generators in the electronic Gel’fand basis, afford a versatile method for the direct calculation of one- and two-body density matrices in the unitary group approach framework.

A uniformly coupled double-quantum Hamiltonian for a spin chain has recently been implemented experimentally. We propose a method for the determination of initial quantum states that will provide perfect or near-perfect state transmission for an arbitrary Hamiltonian including this one. By calculating the eigenvalues and eigenvectors of a unitary operator obtained from the free evolution plus an exchange operator, we find that the double-quantum Hamiltonian spin chain will support a three-spin initial encoding that will transfer along the chain with remarkably high fidelity. The fidelity is also found to decrease very slowly with increasing chain length. In addition, we are able to explain previous results showing exceptional transfer using this method.

We discuss a new analytical approach to real-time evolution in quantum many-body systems. Our approach extends the framework of continuous unitary transformations such that it amounts to a novel solution method for the Heisenberg equations of motion for an operator. It is our purpose to illustrate the accuracy of this approach by studying dissipative quantum systems on all timescales. In particular, we obtain results for non-equilibrium correlation functions for general initial conditions. We illustrate our ideas for the exactly solvable dissipative oscillator and, as a non-trivial model, for the dissipative two-state system.