By the method of algebraic dynamics,the exact solutions and Cartan invariant operators of the SU(3) linear nonautonomous quantum system have been obtained,the quantum-classical correspondence of the solutions established.The nonadiabatic Berry phase for a periodic system is also- calculated.
The exact solution and the invariant Cartan operator of the linear nonautono-mous system with SU(2) dynamical group are obtained with the method of algebraic dynamics. The profound quantum-classical correspondence of the solutions is exhibited clearly: the quantum solutions of the system are determined by that of the corresponding classical equations. The exact solutions are employed to calculate the inversion probability of a neutrino in a time-dependent magnetic field and the-particle spin polarization in beam dynamics. Nonadiabatic Berry's phase for a periodic system is also calculated.
A formalism is developed for calculating Berry phases for nonadiabatic time-periodic quantum systems when a dynamical invariant is known. It is found that, when the invariant is periodic and has a nondegenerate spectrum, this method allows a convenient way to obtain generalized Berry phases and the proper cyclic initial states. The method is applied to the generalized harmonic oscillator and the two-level system, where the invariant operators are explicitly constructed. Formulae for the conventional Berry's phases are readily obtained by taking the adiabatic limit of the exact results.
We study the time evolution of quantum systems with a time-dependent Hamiltonian given by a linear combination of SU(1,1) and SU(2) generators. The invariant Hermitian operator is constructed in the same manner as for both the SU(1,1) and SU(2) systems. With the help of the invariant Hermitian operator we obtain not only the exact solutions of the Schr\"odinger equation but also the time-evolution operator. The adiabatic and nonadiabatic Berry phases are calculated with the exact solutions. \textcopyright{} 1996 The American Physical Society.
A convenient method to exactly solve the quantum-nonautonomous systems with non-Hermitian Hamiltonians is proposed. It is shown that a nonadiabatic complete biorthonormal set can be easily obtained by the gauge transformation method in which the algebraic structure of systems has been used. The nonunitary evolution operator is also found by choosing a special gauge function. All auxiliary parameters introduced in the present approach are only determined by some algebraic equations. The dynamics of two quantum-nonautonomous systems ruled by non-Hermitian Hamiltonians, including a two-photon ionization process involving two-state only and a mesoscopic RLC circuit with a source, are treated as the demonstration of our general approach.
A convenient method to exactly solve the quantum-nonautonomous systems with non-Hermitian Hamiltonians is proposed.It is shown that a nonadiabatic complete biorthonormal set can be easily obtained by the gauge transformation method in which the algebraic structure of systems has been used.The nonuitary evolution operator is also found by choosing a special gauge function.All auxiliry parameters introduced in the present approach are only determined by some algebraic equations.The dynamics of two quantum-nonautonomous systems ruled by non-Hermitian Hamiltonians,including a two-photon ionization process involving two-state only and a mesoscopic RLC circuit with a source,are treated as the demonstration of our general approach.
The nonadiabatic Berry's phase is investigated for a quantum system with a dynamical semisimple Lie group within the framework of the generalized cranking approach. An expression for nonadiabatic Berry's phase is given, which indicates that the nonadiabatic Berry's phase is related to the expectation value of Cartan operators along the cranking direction in group space, and that it depends on (i) the geometry of the group space, (ii) the time evolution ray generated by the Hamiltonian (i.e., by the dynamics) in some irreducible representation Hilbert space, and (iii) the cranking rate. The expression also provides a simple algorithm for calculating the nonadiabatic Berry's phase. The general formalism is illustrated by examples of SU(2) dynamic group.