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The linearization of the non-Hermitian Lindblad equation and its applicaton

Jian Xu, You-neng Guo

Published Apr 7, 2020• 0 citations
OpenAlex

Abstract

Non-Hermiticity and dissipations are two different effects of open systems and the interaction between them has yet to be studied detailedly. In this paper, we theoretically and numerically explore an open system with non-Hermiticity and dissipations. We solve non-Hermitian(NH) Hamiltonians combined with dissipative channels by using the vectorized Lindblad equation, which is an exactly solvable method to address open systems with dissipations. According to this frame and taking a two-level system as an example, we predict a new class of steady states where one eigenvalue of this system is real and the others are complex and a new class of exceptional points(EPs) due to the interplay between non-Hermiticity and dissipations. Further, we reveal the topological properties of eigenstates with $Re[\lambda]=0$ regime, e.g NH Fermi arcs. In short, our work brings a new insight into NH systems and provides a way to study open systems with non-Hermiticity and dissipations.

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Non-Hermiticity and dissipations are two different effects of open systems and the interaction between them has yet to be studied detailedly. In this paper, we theoretically and numerically explore an open system with non-Hermiticity and dissipations. We solve non-Hermitian(NH) Hamiltonians combined with dissipative channels by using the vectorized Lindblad equation, which is an exactly solvable method to address open systems with dissipations. According to this frame and taking a two-level system as an example, we predict a new class of steady states where one eigenvalue of this system is real and the others are complex and a new class of exceptional points(EPs) due to the interplay between non-Hermiticity and dissipations. Further, we reveal the topological properties of eigenstates with Re[λ]=0Re[\lambda]=0Re[λ]=0 regime, e.g NH Fermi arcs. In short, our work brings a new insight into NH systems and provides a way to study open systems with non-Hermiticity and dissipations.
Abstract Decoherence and non-Hermiticity are two different effects of the open quantum systems. Both of them have triggered many interesting phenomena. In this paper, we theoretically study an open two-level non-Hermitian system coupling to a dissipative environment by solving the vectorized Lindblad equation. This scheme provides us a powerful framework to address widespread open systems with gain, loss and dissipation. Our results show that there exist a new class of exceptional points and steady states due to the interplay between non-Hermiticity and decoherence. Furthermore, we also demonstrate a new-type topological properties of eigenstates with zero real-part of eigenvalues (Re[λ] = 0) which are corresponding to Fermi arcs. It is revealed that the phases of eigenstates located in Fermi arcs regime have a topological phase |π/2| which is totally unaffected by the dissipative environment. Our results provide a promising approach for further uncovering and understanding the intriguing properties of non-Hermitian open systems.
The open quantum system can be described by either a Lindblad master equation or a non-Hermitian Hamiltonian (NHH). However, these two descriptions usually have different exceptional points (EPs), associated with the degeneracies in the open quantum system. Here, considering a dissipative quantum Rabi model, we study the spectral features of EPs in these two descriptions and explore their connections. We find that, although the EPs in these two descriptions are usually different, the EPs of NHH will be consistent with the EPs of master equation in the weak coupling regime. Further, we find that the quantum Fisher information (QFI), which measures the statistical distance between quantum states, can be used as a signature for the appearance of EPs. Our study may give a theoretical guidance for exploring the properties of EPs in open quantum systems.
For decades, Hermiticity was considered an immutable axiom of quantum mechanics, essential for ensuring real energies and unitary evolution. This perspective has shifted radically, driven by the realization that non-Hermitian Hamiltonians provide a powerful effective description of open quantum systems, granting access to unique phenomena such as Exceptional Points and the Non-Hermitian Skin Effect. In this Perspective, we chart the trajectory of this field, moving from its established foundations in single-particle, linear models to the emerging frontier of interacting many-body systems. We first clarify the physical origins of non-Hermitian dynamics, distinguishing between mean-field approximations, conditional "no-click" evolution, and exact Liouvillian dynamics. We then focus on the rich phenomenology arising from the interplay of non-Hermiticity and interactions. We discuss interaction-induced topological phases, the generalization of skin effects to the many-body Hilbert space, and the distinct signatures of dissipative quantum chaos and complexity. Finally, we highlight collective phenomena in nonlinear regimes, including skin solitons and dissipative phase transitions. We also comment on measurement-induced entanglement transitions and their relation to non-Hermitian spectra and topology. By synthesizing these diverse developments, we provide a roadmap for the future of non-Hermitian physics.
We study two-body non-Hermitian physics in the context of an open dissipative system depicted by the Lindblad master equation. Adopting a minimal lattice model of a handful of interacting fermions with single-particle dissipation, we show that the non-Hermitian effective Hamiltonian of the master equation gives rise to two-body scattering states with state- and interaction-dependent parity–time transition. The resulting two-body exceptional points can be extracted from the trace-preserving density-matrix dynamics of the same dissipative system with three atoms. Our results not only demonstrate the interplay of parity-time symmetry and interaction on the exact few-body level, but also serve as a minimal illustration on how key features of non-Hermitian few-body physics can be probed in an open dissipative many-body system.
We review analyses of open quantum systems. We show how non-Hermiticity arises in an open quantum system with an infinite environment, focusing on the one-body problem. One of the reasons for taking the present approach is that we can solve the problem completely, making it easier to see the structures of problems involving open quantum systems. We show that this results in the discovery of a new complete set, which is one of the main topics of the present article. Another reason for focusing on the one-body problem is that the theory permits the strong coupling between the system and the environment. In the current research landscape, it is valuable to revisit the one-body problem for open quantum systems, which can be solved accurately for arbitrary strengths of the system-environment couplings. A rigorous understanding of the problem structures in the present approach will be helpful when we tackle problems with many-body interactions. First, we consider potential scattering and directly define the resonant state as an eigenstate of the Schrödinger equation under the Siegert outgoing boundary condition. We show that the resonant eigenstate can have a complex energy eigenvalue, even though the Hamiltonian is seemingly Hermitian. Second, we introduce the Feshbach formalism, which eliminates the infinite degrees of freedom of the environment and represents its effect as a complex potential. The resulting effective Hamiltonian is explicitly non-Hermitian. By unifying these two ways of defining resonant states, we obtain a new complete set of bases for the scattering problem that contains all discrete eigenstates, including resonant states. We finally mention the non-Markovian dynamics of open quantum systems. We emphasize the time-reversal symmetry of the dynamics that continuously connects the past and the future. We can capture it using the new complete set that we develop here.
We compare two approaches to open quantum systems, namely, the non-Hermitian dynamics and the Lindblad master equation. In order to deal with more general dissipative phenomena, we propose the unified master equation that combines the characteristics of both of these approaches. This allows us to assess the differences between them as well as to clarify which observed features come from the Lindblad or the non-Hermitian part, when it comes to experiment. Using a generic two-mode single-atom laser system as a practical example, we analytically solve the dynamics of the normalized density matrix operator. We study the two-level model in a number of cases (depending on parameters and types of dynamics), compute different observables and study their physical properties. It turns out that one can not only able to describe the different types of damping in dissipative quantum optical systems but also mimic the undamped anharmonic oscillatory phenomena which happen in quantum systems with more than two levels (while staying within the framework of the analytically simple two-mode approximation).