Ballistic diffusion versus damped oscillation of energy in a <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>P</mml:mi> <mml:mi>T</mml:mi> </mml:math> -symmetric quantum kicked harmonic oscillator

Abstract We numerically study the quantum dynamics of a <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi class="MJX-tex-calligraphic" mathvariant="script">P</mml:mi> <mml:mi class="MJX-tex-calligraphic" mathvariant="script">T</mml:mi> </mml:math> -symmetric kicked harmonic oscillator. The <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi class="MJX-tex-calligraphic" mathvariant="script">P</mml:mi> <mml:mi class="MJX-tex-calligraphic" mathvariant="script">T</mml:mi> </mml:math> symmetry breaks spontaneously when the non-Hermitian driving strength surpasses the exceptional point. In the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi class="MJX-tex-calligraphic" mathvariant="script">P</mml:mi> <mml:mi class="MJX-tex-calligraphic" mathvariant="script">T</mml:mi> </mml:math> -broken phase, we observe that directed current of momentum and ballistic diffusion of energy coexist under the non-resonant conditions, whereas both the momentum and energy oscillate as damped cosine functions with identical frequencies under the resonant conditions. The research shows that the directed current of momentum and ballistic diffusion of energy arise from nearest-neighbor hopping between momentum eigenstates with the non-Hermitian driving, while the damped oscillations of momentum and energy originate from resonant coupling between the non-Hermitian driving and the harmonic oscillator. Our findings indicate that the non-Hermiticity and the frequency characteristic of this system collectively result in these distinctive dynamical behaviors.