Effective Hamiltonian in Algebraic Approach

Qing-Wu Wang

Using algebraic dynamics and matrix diagnization,the geometric phase of a non-symmetry 3-particle spin chain has been calculated.The dependence of the geometric phase of this system on the algebraic structure of Hamiltonian is studied.It is found that when the algebraic structure of Hamiltonian is broken,a drastic change in the structure of the geometric phase takes place.

Jan Peřina

J. L. Gruver, J. Aliaga

We show that the Stark and Kerr Hamiltonians are deeply connected in the framework of dynamical algebras. We found that the algebras for both Hamiltonians are the same when physically relevant magnitudes, such as the population inversion and the $n$th order coherence function are considered as elements of a Lie algebra under commutation with the Hamiltonians. By analyzing the equations of motion we were able to find a set of conditions for the characteristic magnitudes of both Hamiltonians that leads to the same dynamical behavior for all the elements of the group. Finally, we conclude that the results of this paper can be generalized to any extension of the Jaynes-Cummings model, where any of the different elements of the group are present.

忍 永田, 敏行 室田, 弘治 坂東

Ángel Ballesteros, Romina Ramírez, M. Reboiro

In this work, $\mathcal{PT}$-symmetric Hamiltonians defined on quantum $sl(2, \mathbb R)$ algebras are presented. We study the spectrum of a family of non-Hermitian Hamiltonians written in terms of the generators of the non-standard $U_{z}(sl(2, \mathbb R))$ Hopf algebra deformation of $sl(2, \mathbb R)$. By making use of a particular boson representation of the generators of $U_{z}(sl(2, \mathbb R))$, both the co-product and the commutation relations of the quantum algebra are shown to be invariant under the $\mathcal{PT}$-transformation. In terms of these operators, we construct several finite dimensional $\mathcal{PT}$-symmetry Hamiltonians, whose spectrum is analytically obtained for any arbitrary dimension. In particular, we show the appearance of Exceptional Points in the space of model parameters and we discuss the behaviour of the spectrum both in the exact $\mathcal{PT}$-symmetry and the broken $\mathcal{PT}$-symmetry dynamical phases. As an application, we show that this non-standard quantum algebra can be used to define an effective model Hamiltonian describing accurately the experimental spectra of three-electron hybrid qubits based on asymmetric double quantum dots. Remarkably enough, in this effective model, the deformation parameter $z$ has to be identified with the detuning parameter of the system.