Lowest Weight Representations, Singular Vectors and Invariant Equations for a Class of Conformal Galilei Algebras

The conformal Galilei algebra (CGA) is a non-semisimple Lie algebra labelled by two parameters $d$ and $\ell$. The aim of the present work is to investigate the lowest weight representations of CGA with $d = 1$ for any integer value of $\ell$. First we focus on the reducibility of the Verma modules. We give a formula for the Shapovalov determinant and it follows that the Verma module is irreducible if $\ell = 1$ and the lowest weight is nonvanishing. We prove that the Verma modules contain many singular vectors, i.e., they are reducible when $\ell \neq 1$. Using the singular vectors, hierarchies of partial differential equations defined on the group manifold are derived. The differential equations are invariant under the kinematical transformation generated by CGA. Finally we construct irreducible lowest weight modules obtained from the reducible Verma modules

Enhanced quantum phase synchronization of a laser-driven qubit under non-Markovian dynamics

In this paper, we demonstrate transient dynamics of Husimi Q-representation to visualize and characterize the phase synchronization behavior of a two-level system (qubit) driven by a laser field in both the Markov and non-Markov regime. In the Markov regime, phase preference of the qubit goes away in the long time limit, whereas the long-time phase localization persists in the non-Markovian regime. We also plot the maximum of the shifted phase distribution in two different ways: (a) by varying the detuning and laser drive strength, and (b) by varying the system-bath coupling and laser drive strength. Signature of quantum phase synchronization viz. the Arnold tongue is demonstrated through the maximal value of the shifted phase distribution. The phase synchronization is observed inside the tongue region while the region outside the tongue is desynchronized. The synchronization regions are determined by various system-environment parameters and the qubit phase synchronization is shown to be enhanced in the non-Markov regime.Comment: 6 figure