Emergence of PT-symmetry breaking in open quantum systems

Abstract

The effect of \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 in coupled systems with balanced gain and loss has recently attracted considerable attention and has been demonstrated in various photonic, electrical and mechanical systems in the classical regime. However, it is still an unsolved problem how to generalize the concept of \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 to the quantum domain, where the conventional definition in terms of non-Hermitian Hamiltonians is not applicable. Here we introduce a symmetry relation for Liouville operators that describe the dissipative evolution of arbitrary open quantum systems. Specifically, we show that the invariance of the Liouvillian under this symmetry transformation implies the existence of stationary states with preserved and broken parity symmetry. As the dimension of the Hilbert space grows, the transition between these two limiting phases becomes increasingly sharp and the classically expected \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 transition is recovered. This quantum-to-classical correspondence allows us to establish a common theoretical framework to identify and accurately describe \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 effects in a large variety of physical systems, operated both in the classical and quantum regimes.

Topics & Concepts