Symmetries of Liouvillians of squeeze-driven parametric oscillators

TL;DR

This paper explores the symmetries of Liouvillian superoperators in squeeze-driven parametric oscillators, revealing an $su(2)$ symmetry at specific parameter ratios and analyzing phase transitions and temperature effects relevant to quantum computing.

Contribution

It uncovers a quasi-spin $su(2)$ symmetry in the Liouvillian of squeeze-driven Kerr oscillators at integer parameter ratios, extending symmetry understanding from Hamiltonians to Liouvillians.

Findings

Identified $su(2)$ symmetry at integer $rac{ ext{detuning}}{K}$ ratios.

Characterized the double-ellipsoidal structure of the Liouvillian in $su(2)$ representations.

Analyzed phase transitions and temperature effects on the Liouvillian spectrum.

Abstract

We study the symmetries of the Liouville superoperator of one dimensional parametric oscillators, especially the so-called squeeze-driven Kerr oscillator, and discover a remarkable quasi-spin symmetry $su(2)$ at integer values of the ratio $\eta =\omega /K$ of the detuning parameter $\omega$ to the Kerr coefficient $K$, which reflects the symmetry previously found for the Hamiltonian operator. We find that the Liouvillian of an $su(2)$ representation $\left\vert j,m_{j}\right\rangle$ has a characteristic double-ellipsoidal structure, and calculate the relaxation time $T_{X}$ for this structure. We then study the phase transitions of the Liouvillian which occur as a function of the parameters $\xi =\varepsilon _{2}/K$ and $\eta=\omega /K$. Finally, we study the temperature dependence of the spectrum of eigenvalues of the Liouvillian. Our findings may have applications in the generation and stabilization of states of interest in quantum computing.