Algebraic methods and exact solutions of quantum parametric oscillators

Şirin A. Büyükaşık, Zehra Çayiç

We introduce exactly solvable quantum parametric oscillators, which are generalizations of the quantum problems related with the classical orthogonal polynomials of Hermite, Laguerre, and Jacobi type, introduced in the work of Büyükaşık et al. [J. Math. Phys. 50, 072102 (2009)]. Quantization of these models with specific damping, frequency, and external forces is obtained using the Wei-Norman Lie algebraic approach. This determines the evolution operator exactly in terms of two linearly independent homogeneous solutions and a particular solution of the corresponding classical equation of motion. Then, time-evolution of wave functions and coherent states are found explicitly. Probability densities, expectation values, and uncertainty relations are evaluated and their properties are investigated under the influence of the external terms.

Paul Kinsler

В. С. Попов, A. M. Perelomov

Hiroaki Mishima, Yuki Izumida

The transitionless tracking (TT) algorithm enables the exact tracking of quantum adiabatic dynamics in an arbitrary short time by adding a counterdiabatic Hamiltonian to the original adiabatic Hamiltonian. By applying Husimi's method originally developed for a quantum parametric oscillator (QPO) to the transitionless QPO achieved using the TT algorithm, we obtain the transition probability generating function with a time-dependent parameter constituted with solutions of the corresponding classical parametric oscillator (CPO). By obtaining the explicit solutions of this CPO using the phase-amplitude method, we find that the time-dependent parameter can be reduced to the frequency ratio between the Hamiltonians without and with the counterdiabatic Hamiltonian, from which we can easily characterize the result achieved by the TT algorithm. We illustrate our theory by showing the trajectories of the CPO on the classical phase space, which elucidate the effect of the counterdiabatic Hamiltonian of the QPO.

pD M Tibaduizap

В. С. Попов, A. M. Perelomov