Abstract In this work, we show that in the regime of real and non-degenerate spectrum, the quasi-Hermitian framework provides an effective description of the conditional no-jump evolution of certain Markovian open systems within invariant subspaces while the full Lindblad dynamics governs the unconditional dynamics which includes quantum jumps. We present a three-site fermionic model with asymmetric hopping amplitudes whose single-particle generator is intrinsically non-Hermitian yet supports an all-real spectrum in a finite region of the parameter space. We construct an explicit Lindbladian-based embedding in which the same Hamiltonian arises as the no-jump generator of a completely positive and trace-preserving Lindblad equation, up to a particle-decay term. Due to particle-number conservation, this shift reduces to a scalar within each fixed-particle-number sector. We further analyze the single-particle spectrum and identify a third-order exceptional point that lies on the boundary marking the breakdown of quasi-hermiticity in a given fixed-particle-number sector.
Abstract In this work, we show that in the regime of real and non-degenerate spectrum, the quasi-Hermitian framework provides an effective description of the conditional no-jump evolution of certain Markovian open systems within invariant subspaces while the full Lindblad dynamics governs the unconditional dynamics which includes quantum jumps. We present a three-site fermionic model with asymmetric hopping amplitudes whose single-particle generator is intrinsically non-Hermitian yet supports an all-real spectrum in a finite region of the parameter space. We construct an explicit Lindbladian-based embedding in which the same Hamiltonian arises as the no-jump generator of a completely positive and trace-preserving Lindblad equation, up to a particle-decay term. Due to particle-number conservation, this shift reduces to a scalar within each fixed-particle-number sector. We further analyze the single-particle spectrum and identify a third-order exceptional point that lies on the boundary marking the breakdown of quasi-hermiticity in a given fixed-particle-number sector.
We present a general proof that non-Hermitian dynamics and Lindblad dynamics with only decay terms are equivalent in the highest particle subspace. We then propose an unbiased method to determine if a system's dynamics in the highest-particle subspace is non-Hermitian. We exemplify this for a simple two-site decay system connected to two baths, and find that the exact solution is well approximated by non-Hermitian dynamics only in the weak-coupling and in the singular-coupling limits, where a Lindbladian description was already known to be accurate. The fact that an accurate non-Hermitian description is so limited, even for such a simple system, raises doubts about how valid such descriptions are for more complicated systems away from these asymptotic limits. Finally, we prove that for models with a nondegenerate system Hamiltonian, exceptional points cannot occur in the weak-coupling limit. This result is relevant for the design of experiments that aim to identify such exceptional points.
We investigate the relation between non-Hermitian Hamiltonian and Lindblad dynamics in nonequilibrium open quantum systems. Non-Hermitian models can extend phase diagrams and enable sensing advantages, but such effects often rely on postselection, raising questions about their relevance for unconditional dynamics. Using a minimal two-qubit setup mediating a heat current, we compare local and global Markovian master equations with their non-Hermitian counterparts. We observe that exceptional points emerge only in the local master equation and in the corresponding non-Hermitian Hamiltonian at sufficiently strong nonequilibrium. We further consider hybrid configurations, where one bath is treated with a Lindblad description and the other with a non-Hermitian approach, interpolating between the two extremes. Our results contribute understanding the role of quantum jumps and exceptional points in nonequilibrium open quantum systems and identify a simple, experimentally accessible architecture, realizable, for instance, in circuit-QED platforms, for their exploration.
The Lindblad master equation (LME) describing the Markovian dynamics of the quantum open system can be understood as the evolution of the effective non-Hermitian Hamiltonian balanced with random quantum jumps. Here we investigate the balance-breaking dynamics by partly eliminating jumps from postselection experiments. To describe this dynamics, a nonlinear Lindblad master equation (NLME) is derived from the quantum trajectory method. However, the NLME shows significant advantages in analytical analysis over the quantum trajectory method. Using the NLME, we classify the dynamics into two classes. In the trivial class, the process of reducing jumps is completely equivalent to weakening the coupling from the environment. In contrast, the nontrivial class presents more complex dynamics. We study a prototypical model within this class and demonstrate the existence of the postselected skin effect whose steady state is characterized by the accumulation of particles on one side. The steady-state distribution can be fitted by a scale-invariant tanh function which is different from the uniform distribution of LME. Furthermore, the NLME can give a reasonable framework for studying the interplay and competition between the non-Hermitian Hamiltonians and dissipative terms. We show this by capturing the characteristics of the trajectory-averaged entanglement entropy influenced by non-Hermitian skin effect and Zeno effect in the model of postselected skin effect.
Phenomenological non-Hermitian Hamiltonians track selected signatures of complex reservoir dynamics, while post-selected no-jump effective Hamiltonians derived from microscopic open-system theory reveal the underlying system--reservoir physics. We derive such a Hamiltonian for the open Jaynes--Cummings model using a Moore--Penrose normalized su(2) representation that removes the vacuum-sector singularity and diagonalizes the full Hamiltonian by one operator rotation. Starting from a zero-temperature bosonic reservoir, we obtain a Gorini--Kossakowski--Sudarshan--Lindblad master equation under the Born--Markov approximation with full Bohr-frequency resolution. We use partial Bohr-frequency resolution to build a consistent post-selected no-jump Hamiltonian near exceptional points, where decay rates become comparable to Rabi frequencies and remove the scale separation behind full resolution. The normalized su(2) form of the resulting non-Hermitian Jaynes--Cummings Hamiltonian reveals the effects of Lamb-shifted detuning, diagonal loss imbalance, and reservoir-modified coupling. Our microscopic exceptional-point analysis recovers the experimentally reported single-excitation exceptional point for unequal independent losses and identifies regimes absent from the standard phenomenological model; for example, equal correlated losses with orthogonal channel phase produce a second-order exceptional point at the same loss-to-coupling ratio in every excitation sector.
Non-Hermitian Hamiltonians and Lindblad operators are some of the most important generators of dynamics for describing quantum systems interacting with different kinds of environments. The first type differs from conservative evolution by an anti-Hermitian term that causes particle decay, while the second type differs by a dissipation operator in Lindblad form that allows energy exchange with a bath. However, although under some conditions the two types of maps can be used to describe the same observable, they form a disjoint set. In this work, we propose a generalized generator of dynamics of the form Lmixed(z,ρS)=−i[H,ρS]+∑i(z+Γc,iΓc,iFiρSFi†−21{Fi†Fi,ρS}+) that depends on a general energy z, and has a tunable parameter Γc that determines the degree of particle density lost. It has as its limits non-Hermitian (Γc→0) and Lindbladian dynamics (Γc→∞). The intermediate regime evolves density matrices such that 0≤Tr(ρS)≤1. We derive our generator with the help of an ancillary continuum manifold acting as a sink for particle density. The evolution describes a system that can exchange both particle density and energy with its environment. We illustrate its features for a two level system and a five M level system with a coherent population trapping point.
Abstract The dynamics of Markovian open quantum systems are described by Lindblad master equations. For fermionic and bosonic systems that are quasi-free, i.e. with Hamiltonians that are quadratic in the ladder operators and Lindblad operators that are linear in the ladder operators, we derive the equation of motion for the covariance matrix. This determines the evolution of Gaussian initial states and the steady states, which are also Gaussian. Using ladder super-operators (a.k.a. third quantization), we show how the Liouvillian can be transformed to a many-body Jordan normal form which also reveals the full many-body spectrum. Extending previous work by Prosen and Seligman, we treat fermionic and bosonic systems on equal footing with Majorana operators, shorten and complete some derivations, also address the odd-parity sector for fermions, give a criterion for the existence of bosonic steady states, cover non-diagonalizable Liouvillians also for bosons, and include quadratic systems. In extension of the quasi-free open systems, quadratic open systems comprise additional Hermitian Lindblad operators that are quadratic in the ladder operators. While Gaussian states may then evolve into non-Gaussian states, the Liouvillian can still be transformed to a useful block-triangular form, and the equations of motion for k -point Green’s functions form a closed hierarchy. Based on this formalism, results on criticality and dissipative phase transitions in such models are discussed in a companion paper.