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Physical Review A

Role of nonorthogonality of energy eigenstates in quantum systems with localized losses

Jan Wiersig

Published Nov 6, 2018• 9 citations
Publisher

Abstract

The dynamics of a wave in an open quantum system can be described by the eigenstates of the non-Hermitian Hamiltonian, provided that it is not at an exceptional point. Naively, one would expect that the decay of the energy eigenstates directly determines the decay of the total wave intensity. We show, however, that this is not the case for systems with localized losses. Instead, the nonorthogonality of the energy eigenstates plays an important role. It can to a large extent compensate for the initial decay of the eigenstates leading to a transient power-law decay of the wave intensity with a potentially large exponent.

DOI
10.1103/physreva.98.052105

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The dynamics of a wave in an open quantum system can be described by the eigenstates of the non-Hermitian Hamiltonian, provided that it is not at an exceptional point. Naively, one would expect that the decay of the energy eigenstates directly determines the decay of the total wave intensity. We show, however, that this is not the case for systems with localized losses. Instead, the nonorthogonality of the energy eigenstates plays an important role. It can to a large extent compensate for the initial decay of the eigenstates leading to a transient power-law decay of the wave intensity with a potentially large exponent.
Non-Hermitian quantum physics is used successfully for the description of different puzzling experimental results, which are observed in open quantum systems. Mostly, the influence of exceptional points on the dynamical properties of the system is studied. At these points, two complex eigenvalues Ei≡Ei+iΓi/2 of the non-Hermitian Hamiltonian H coalesce (where Ei is the energy and Γi is the inverse lifetime of the state i ). We show that also the eigenfunctions Φi of the two states play an important role, sometimes even the dominant one. Besides exceptional points, other critical points exist in non-Hermitian quantum physics. At these points a=acr in the parameter space, the biorthogonal eigenfunctions of H become orthogonal. For illustration, we show characteristic numerical results.
Non-Hermitian Hamiltonians with nonreciprocal coupling can achieve amplification of initial states without external gain due to a kind of inherent source. We discuss the source and its effect on time evolution in terms of complex eigenenergies and nonorthogonal eigenstates. Demonstrating two extreme cases of Hamiltonians, namely one having complex eigenenergies with orthogonal eigenstates and one having real eigenenergies with nonorthogonal eigenstates, we elucidate the differences between the amplifications from complex eigenenergies and from nonorthogonal eigenstates.
Abstract Over the past two decades, open systems that are described by a non-Hermitian Hamiltonian have become a subject of intense research. These systems encompass classical wave systems with balanced gain and loss, semiclassical models with mode selective losses, and minimal quantum systems, and the meteoric research on them has mainly focused on the wide range of novel functionalities they demonstrate. Here, we address the following questions: Does anything remain constant in the dynamics of such open systems? What are the consequences of such conserved quantities? Through spectral-decomposition method and explicit, recursive procedure, we obtain all conserved observables for general PT -symmetric systems. We then generalize the analysis to Hamiltonians with other antilinear symmetries, and discuss the consequences of conservation laws for open systems. We illustrate our findings with several physically motivated examples.
Nonorthogonality of eigenstates is a fundamental characteristic of non-Hermitian systems. It is expected that nonorthogonality-induced effects could be enhanced in the vicinity of the exceptional points (EPs) where the multiple eigenstates coalesce into one. Here, focusing on the wave-packet motions in non-Hermitian systems, we find that a wave packet near the EPs could exhibit new types of Zitterbewegung- (ZB-) like motion arising from the nonorthogonality of eigenstates. The observed ZB-like motion of a wave packet centered around a single eigenstate contradicts the notion that ZB motion only occurs for a wave packet centered around a superposed state. We further show the existence of a ZB-like motion in the form of an interesting helical trajectory of the wave packet on the incident plane, stemming from the imaginary part of group velocities. These exotic ZB-like motions may provide great insight into nonorthogonal eigenstates and complex eigenfrequencies of non-Hermitian systems.
In open quantum systems where the effective Hamiltonian is not Hermitian, it is known that the adiabatic (or instantaneous) basis can be multivalued: by adiabatically transporting an eigenstate along a closed loop in the parameter space of the Hamiltonian, it is possible to end up in an eigenstate different from the initial eigenstate. This ‘adiabatic flip’ effect is an outcome of the appearance of a degeneracy known as an ‘exceptional point’ inside the loop. We show that contrary to what is expected of the transport properties of the eigenstate basis, the interplay between gain/loss and non-adiabatic couplings imposes fundamental limitations on the observability of this adiabatic flip effect.
One of the unique features of non-Hermitian Hamiltonians is the non-Hermitian skin effect, namely, that the eigenstates are exponentially localized at the boundary of the system. For open quantum systems, a short-time evolution can often be well described by the effective non-Hermitian Hamiltonians, while long-time dynamics calls for the Lindblad master equations, in which the Liouvillian superoperators generate time evolution. In this Letter, we find that Liouvillian superoperators can exhibit the non-Hermitian skin effect, and uncover its unexpected physical consequences. It is shown that the non-Hermitian skin effect dramatically shapes the long-time dynamics, such that the damping in a class of open quantum systems is algebraic under periodic boundary conditions but exponential under open boundary conditions. Moreover, the non-Hermitian skin effect and non-Bloch bands cause a chiral damping with a sharp wave front. These phenomena are beyond the effective non-Hermitian Hamiltonians; instead, they belong to the non-Hermitian physics of full-fledged open quantum dynamics.