Signature of exceptional point phase transition in Hermitian systems

T. T. Sergeev1,2,3, A. A. Zyablovsky1,2,3,4, E. S. Andrianov1,2,3, and Yu. E. Lozovik5,6

1Dukhov Research Institute of Automatics, 127055, 22 Sushchevskaya, Moscow Russia
2Moscow Institute of Physics and Technology, 141700, 9 Institutskiy pereulok, Dolgoprudny Moscow region, Russia
3Institute for Theoretical and Applied Electromagnetics, 125412, 13 Izhorskaya, Moscow Russia
4Kotelnikov Institute of Radioengineering and Electronics RAS, 125009, 11-7 Mokhovaya, Moscow Russia
5Institute of Spectroscopy Russian Academy of Sciences, 108840, 5 Fizicheskaya, Troitsk, Moscow, Russia
6MIEM at National Research University Higher School of Economics, 101000, 20 Myasnitskaya, Moscow, Russia

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Abstract

Exceptional point (EP) is a spectral singularity in non-Hermitian systems. The passing over the EP leads to a phase transition, which endows the system with unconventional features that find a wide range of applications. However, the need of using the dissipation and amplification limits the possible applications of systems with the EP. In this work, we demonstrate an existence of signature of exceptional point phase transition in Hermitian systems that are free from dissipation and amplification. We consider a composite Hermitian system including both two coupled oscillators and their environment consisting only of several tens of degrees of freedom. We show that the dynamics of such a Hermitian system demonstrate a transition, which occurs at the coupling strength between oscillators corresponding to the EP in the non-Hermitian system. This transition manifests itself even in the non-Markovian regime of the system dynamics in which collapses and revivals of the energy occur. Thus, we demonstrate that the phase transition occurring at the passing over the EP in the non-Hermitian system manifests itself in the Hermitian system at all time. We discuss the experimental scheme to observe the signature of EP phase transition in the non-Markovian regime.

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An interaction of system with an environment causes an energy exchange between them. At times smaller than Poincare return time, the energy exchange leads to relaxation processes in the system. At times less than the return time, the systems interacting with the environment are often considered as non-Hermitian. The eigenstates of non-Hermitian systems are no mutually orthogonal. The point in the space of system parameters, at which some of eigenstates coalesce and their eigenvalues coincide is called an exceptional point (EP) of non-Hermitian system. The passing over the EP is accompanied by qualitative changes in the eigenstates, which is referred to as an EP phase transition. At times greater the return time, the system dynamics demonstrate collapses and revivals, which are due to finite size of the environment. In this case, the non-Hermitian consideration is not suitable and an existence of the EP phase transitions early does not discuss.
We demonstrate an existence of a signature of the EP phase transition at times greater the Poincare return time. We consider a Hermitian system including environment consisting of only a few tens of degrees of freedom. We show that the dynamics of such a Hermitian system demonstrate a signature of the EP phase transition at times both smaller and greater than the return time. This transition occurs at the system parameters corresponding to the EP in the non-Hermitian system. We introduce an order parameter that characterizes the EP phase transition in both the Hermitian and non-Hermitian systems. We propose an experimental scheme to observe the signature of EP phase transition in the Hermitian system at time greater than the return time. Our results extend the concept of the EP phase transition to the Hermitian systems.

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