Symmetries of Liouvillians of squeeze-driven parametric oscillators

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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>η</mml:mi> <mml:mo>=</mml:mo> <mml:mi>ω</mml:mi> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>K</mml:mi> </mml:mrow> </mml:math> of the detuning parameter ω 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mrow> <mml:mo>|</mml:mo> <mml:mi>j</mml:mi> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>m</mml:mi> <mml:mrow> <mml:mi>j</mml:mi> </mml:mrow> </mml:msub> <mml:mo>⟩</mml:mo> </mml:mrow> </mml:mrow> </mml:math> 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>ξ</mml:mi> <mml:mo>=</mml:mo> <mml:msub> <mml:mi>ε</mml:mi> <mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:msub> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>K</mml:mi> </mml:mrow> </mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>η</mml:mi> <mml:mo>=</mml:mo> <mml:mi>ω</mml:mi> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>K</mml:mi> </mml:mrow> </mml:math> . 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.