Adiabatic perturbation theory invariant under the unitary transformations

E. Gozzi

Jeong Ryeol Choi

An adiabatic invariant, which is a conserved quantity, is useful for studying quantum and classical properties of dynamical systems. Adiabatic invariants for time-dependent superconducting qubit-oscillator systems and resonators are investigated using the Liouville-von Neumann equation. At first, we derive an invariant for a simple superconducting qubit-oscillator through the introduction of its reduced Hamiltonian. Afterwards, an adiabatic invariant for a nanomechanical resonator linearly interfaced with a superconducting circuit, via a coupling with a time-dependent strength, is evaluated using the technique of unitary transformation. The accuracy of conservation for such invariant quantities is represented in detail. Based on the results of our developments in this paper, perturbation theory is applicable to the research of quantum characteristics of more complicated qubit systems that are described by a time-dependent Hamiltonian involving nonlinear terms.

Jian Bo Fu, Xinhua Li, Xiao-Chun Gao +1 more

A new formulation is proposed for the investigation of the non-adiabatic non-Abelian connection and phase factor by making use of the invariant theory and invariant-related unitary transformation method. In order to illustrate the formulation, we calculate the non-adiabatic non-Abelian connection and phase factor associated with the motion of a charged particle in a time-dependent magnetic field (the time-dependent Landau level problem).

L. M. Garrido

Xu Hai-Lei, Shen Jian-Qi, Chen Yi-Xin +1 more

The perturbative evaluation of the time-dependent quantum transport is investigated by making use of the Lewis-Riesenfeld invariant theory and the invariant-related unitary transformation formulation in this paper.We obtain the complete set of solutions of the time-dependent Schrdinger equation governing the behaviour of electrons in a double-well structure. Based on these exact solutions,we obtain the first-order quantum rate equation by regarding the interaction Hamiltonian between the double-well structures and the heat reservoirs as the perturbative Hamiltonian.An approach to the geometric phase factor (Berry's phase factor) in the adiabatic quantum transport process is briefly discussed in this paper.We hold that the formulation presented here has advantages in dealing with transport problem over the method proposed by Gurvitz et al.