Phase transitions from heating to non-heating in $SU(1,1)$ quantum dynamics: Applications to Bose-Einstein condensates and periodically driven coupled oscillators

We study the entanglement properties in non-equilibrium quantum systems with the SU(1,1) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>S</mml:mi> <mml:mi>U</mml:mi> <mml:mo stretchy="false" form="prefix">(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false" form="postfix">)</mml:mo> </mml:mrow> </mml:math> structure. Through Möbius transformation, we map the dynamics of these systems following a sudden quench or a periodic drive onto three distinct trajectories on the Poincaré disc, corresponding the heating, non-heating, and a phase boundary describing these non-equilibrium quantum states. We consider two experimentally feasible systems where their quantum dynamics exhibit the SU(1,1) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>S</mml:mi> <mml:mi>U</mml:mi> <mml:mo stretchy="false" form="prefix">(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false" form="postfix">)</mml:mo> </mml:mrow> </mml:math> structure: the quench dynamics of the Bose-Einstein condensates and the periodically driven coupled oscillators. In both cases, the heating, non-heating phases, and their boundary manifest through distinct signatures in the phonon population where exponential, oscillatory, and linear growths classify these phases. Similarly, the entanglement entropy and negativity also exhibit distinct behaviors (linearly, oscillatory, and logarithmic growths) characterizing these phases, respectively. Notibly, for the periodically driven coupled oscillators, the non-equilibrium properties are characterized by two sets of SU(1,1) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>S</mml:mi> <mml:mi>U</mml:mi> <mml:mo stretchy="false" form="prefix">(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false" form="postfix">)</mml:mo> </mml:mrow> </mml:math> generators. The corresponding two sets of the trajectories on two Poincaré discs lead to a more complex phase diagram. We identify two distinct phases within the heating region discernible solely by the growth rate of the entanglement entropy, where a discontinuity is observed when varying the parameters across the phase boundary within in heating region. This discontinuity is not observed in the phonon population.