Quantum chaos in PT symmetric quantum systems

In this study, we explore the interplay between \mathcal{PT} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mstyle mathvariant="script"> <mml:mi>𝒫</mml:mi> <mml:mi>𝒯</mml:mi> </mml:mstyle> </mml:math> -symmetry and quantum chaos in a non-Hermitian dynamical system. We consider an extension of the standard diagnostics of quantum chaos, namely the complex level spacing ratio and out-of-time-ordered correlators (OTOCs), to study the \mathcal{PT} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mstyle mathvariant="script"> <mml:mi>𝒫</mml:mi> <mml:mi>𝒯</mml:mi> </mml:mstyle> </mml:math> -symmetric quantum kicked rotor model. The kicked rotor has long been regarded as a paradigmatic dynamic system to study classical and quantum chaos. By introducing non-Hermiticity in the quantum kicked rotor, we uncover new phases and transitions that are absent in the Hermitian system. From the study of the complex level spacing ratio, we locate three regimes – one which is integrable and \mathcal{PT} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mstyle mathvariant="script"> <mml:mi>𝒫</mml:mi> <mml:mi>𝒯</mml:mi> </mml:mstyle> </mml:math> -symmetry, another which is chaotic with \mathcal{PT} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mstyle mathvariant="script"> <mml:mi>𝒫</mml:mi> <mml:mi>𝒯</mml:mi> </mml:mstyle> </mml:math> -symmetry and a third which is chaotic but with broken \mathcal{PT} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mstyle mathvariant="script"> <mml:mi>𝒫</mml:mi> <mml:mi>𝒯</mml:mi> </mml:mstyle> </mml:math> -symmetry. We find that the complex level spacing ratio can distinguish between all three phases. Since calculations of the OTOC can be related to those of the classical Lyapunov exponent in the semi-classical limit, we investigate its nature in these regimes and at the phase boundaries. In the phases with \mathcal{PT} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mstyle mathvariant="script"> <mml:mi>𝒫</mml:mi> <mml:mi>𝒯</mml:mi> </mml:mstyle> </mml:math> -symmetry, the OTOC exhibits behaviour akin to what is observed in the Hermitian system in both the integrable and chaotic regimes. Moreover, in the \mathcal{PT} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mstyle mathvariant="script"> <mml:mi>𝒫</mml:mi> <mml:mi>𝒯</mml:mi> </mml:mstyle> </mml:math> -symmetry broken phase, the OTOC demonstrates additional exponential growth stemming from the complex nature of the eigenvalue spectrum at later times. We derive the analytical form of the late-time behaviour of the OTOC. By defining a normalized OTOC to mitigate the effects caused by \mathcal{PT} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mstyle mathvariant="script"> <mml:mi>𝒫</mml:mi> <mml:mi>𝒯</mml:mi> </mml:mstyle> </mml:math> -symmetry breaking, we show that the OTOC exhibits singular behaviour at the transition from the \mathcal{PT} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mstyle mathvariant="script"> <mml:mi>𝒫</mml:mi> <mml:mi>𝒯</mml:mi> </mml:mstyle> </mml:math> -symmetric chaotic phase to the \mathcal{PT} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mstyle mathvariant="script"> <mml:mi>𝒫</mml:mi> <mml:mi>𝒯</mml:mi> </mml:mstyle> </mml:math> -symmetry broken, chaotic phase.