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Non-Hermitian Dynamics in Three-Level Systems: A Perturbative Approach for Time-Dependent Hamiltonians

Gui-Xiang La, Yexin Li, Gong-Ping Zheng

Published Feb 28, 2026• 0 citations
Publisher

Abstract

The conventional time-dependent perturbation theory in quantum mechanics is established within the framework of Hermitian Hamiltonians, applicable for describing quantum transitions and associated energy level responses in such systems. However, this theory has fundamental limitations when applied to non-Hermitian systems. Consequently, researchers have systematically extended time-dependent perturbation theory to non-Hermitian systems, establishing a corresponding mature framework. Building on this foundation, this study extends the theory to investigate the transition dynamics induced by non-Hermitian interactions in non-Hermitian Hamiltonian systems. We employ a biorthogonal basis representation for a three-level non-Hermitian system. This work investigates a system comprising an unperturbed static non-Hermitian Hamiltonian and a periodically driven time-dependent perturbation Hamiltonian. Taking the three-level system as a concrete example, we combine analytical methods with numerical simulations to solve and analyze its dynamical evolution equations. These complementary approaches reveal that when system parameters complete a full cycle around an exceptional point, the transitional behavior exhibits specific evolutionary patterns. In this system, quantum transition probabilities exhibit significant asymmetry and non-conservation that depend on the initial and final states, revealing inherent directional characteristics in the dynamical process. Furthermore, for a three-level, periodically driven non-Hermitian system with time-dependent perturbations, this asymmetry is even more pronounced, manifesting as a distinct disparity between forward and reverse transition probabilities. The periodic driving actively amplifies the asymmetry in the transition process. By designing the perturbation spectrum, selective manipulation of specific quantum states can be achieved. Moreover, transition probabilities can be significantly enhanced under resonance conditions, while non-Hermiticity further breaks the system's inherent symmetry, leading to substantial amplification of transitions in a single direction. By precisely tuning the drive frequency, interactions between specific coupling channels can be selectively enhanced or suppressed. The amplification of channel asymmetry by non-Hermitian properties provides a novel mechanism for directional control of quantum states and opens new pathways for realizing related quantum technologies.

DOI
10.3390/e28030268

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The conventional time-dependent perturbation theory in quantum mechanics is established within the framework of Hermitian Hamiltonians, applicable for describing quantum transitions and associated energy level responses in such systems. However, this theory has fundamental limitations when applied to non-Hermitian systems. Consequently, researchers have systematically extended time-dependent perturbation theory to non-Hermitian systems, establishing a corresponding mature framework. Building on this foundation, this study extends the theory to investigate the transition dynamics induced by non-Hermitian interactions in non-Hermitian Hamiltonian systems. We employ a biorthogonal basis representation for a three-level non-Hermitian system. This work investigates a system comprising an unperturbed static non-Hermitian Hamiltonian and a periodically driven time-dependent perturbation Hamiltonian. Taking the three-level system as a concrete example, we combine analytical methods with numerical simulations to solve and analyze its dynamical evolution equations. These complementary approaches reveal that when system parameters complete a full cycle around an exceptional point, the transitional behavior exhibits specific evolutionary patterns. In this system, quantum transition probabilities exhibit significant asymmetry and non-conservation that depend on the initial and final states, revealing inherent directional characteristics in the dynamical process. Furthermore, for a three-level, periodically driven non-Hermitian system with time-dependent perturbations, this asymmetry is even more pronounced, manifesting as a distinct disparity between forward and reverse transition probabilities. The periodic driving actively amplifies the asymmetry in the transition process. By designing the perturbation spectrum, selective manipulation of specific quantum states can be achieved. Moreover, transition probabilities can be significantly enhanced under resonance conditions, while non-Hermiticity further breaks the system's inherent symmetry, leading to substantial amplification of transitions in a single direction. By precisely tuning the drive frequency, interactions between specific coupling channels can be selectively enhanced or suppressed. The amplification of channel asymmetry by non-Hermitian properties provides a novel mechanism for directional control of quantum states and opens new pathways for realizing related quantum technologies.
Nontrivial spectral properties of non-Hermitian systems can give rise to intriguing effects that lack counterparts in Hermitian systems. For instance, when dynamically varying system parameters along a path enclosing an exceptional point (EP), chiral mode conversion occurs. A recent study [Phys. Rev. Lett. 133, 113802 (2024)] demonstrates the achievability of pure adiabatic state transfer by specifically selecting a trajectory in the system parameter space where the corresponding evolution operator exhibits a real spectrum while winding around an EP. However, the intended adiabatic state transfer becomes fragile when taking into account the effect of the nonadiabatic transition. In this work, we propose a scheme for achieving robust and rapid adiabatic state transfer in time-modulated two-level non-Hermitian systems by appropriately modulating the system Hamiltonian and time-evolution trajectory. Numerical simulations confirm that a complete adiabatic transfer can always be achieved even under nonadiabatic conditions after one period for different initialized adiabatic states, and the scheme remains insensitive to moderate fluctuations in control parameters. Therefore, this scheme offers alternative approaches for quantum-state engineering in non-Hermitian systems.
As a paradigm for describing open-quantum-system dynamics, the non-Hermitian Schrödinger equation generally modifies state norms, complicating coherent control over the system's Hilbert space. With a general instantaneous frame spanned by biorthogonal ancillary bases, we develop a biorthogonal quantum-control theory for discrete-variable systems governed by time-dependent non-Hermitian Hamiltonian. In particular, we derive a necessary and sufficient condition for extracting the transitionless pathways that transport parallel in ket and bra states. When this condition is satisfied by a set of projectors constructed with biorthogonal basis states, the non-Hermitian Hamiltonian becomes diagonal in the relevant biorthogonal subspace, enabling arbitrary initial state to evolve as a superposition of controlled transitionless passages. We further identify the parametric constraints under which the transformation linking the initial and final states becomes unitary. As a result, the final system state is automatically normalized even though the transitional dynamics is nonunitary. Our theory is justified by the bidirectional population transfer and a holomonic quantum gate implemented by a non-Hermitian three-level system in ΛΛΛ type. For the population transfer, controlled gain and loss reduce the required coherent pulse area, enabling shorter transfer times than the Hermitian benchmark with the same average Rabi frequency. Our work integrates the universal quantum control (UQC) into the biorthogonal representation, displaying full-rank passage engineering with coherent operations in non-Hermitian systems.
The traditional framework of quantum mechanics and many classical wave phenomena is underpinned by Hermitian operators, which guarantee real eigenvalues corresponding to observable quantities and orthogonal eigenvectors. However, a growing number of physical systems exhibit non-reciprocal interactions, leading to Hamiltonians or system matrices that are inherently non-Hermitian. This paper delves into the spectral theory of such non-reciprocal systems, exploring the unique mathematical and physical characteristics that emerge when the constraint of Hermiticity is relaxed. We discuss phenomena like complex energy eigenvalues, exceptional points where both eigenvalues and eigenvectors coalesce, and the breakdown of standard perturbation theory. The implications of non-Hermitian spectral properties extend to diverse fields, including parity-time (PT) symmetric photonics, active matter, open quantum systems, and even classical wave propagation in active or lossy media. By extending spectral analysis beyond the Hermitian paradigm, we aim to provide a comprehensive overview of the current understanding, challenges, and future directions in characterizing the dynamics and phases of non-reciprocal systems. This exploration is crucial for advancing both fundamental theoretical understanding and practical applications in emerging technologies that leverage tailored non-reciprocity.
In recent years, there have been intensive studies of non-Hermitian physics and parity–time (PT) symmetry due to their fundamental importance in theory and outstanding applications. A distinctive character in PT-symmetric system is phase transition (spontaneous PT-symmetry breaking), i.e. an all-real energy spectrum changes into an all-complex one when the non-Hermitian parameter exceeds a certain threshold. However, the conditions for PT-symmetric system with real energy spectrum to occur are rather restrictive. The generalization of PT-symmetric potentials to wider classes of non-PT-symmetric complex potentials with all-real energy spectra is a currently important endeavor. A simple PT-symmetric two-level Floquet quantum system is now being actively explored, because it holds potential for the realization of non-unitary single-qubit quantum gate. However, studies of the evolution dynamics of non-PT-symmetric two-level non-Hermitian Floquet quantum system are still relatively rare. In this paper, we investigate the non-Hermitian physics of a periodically driven non-PT-symmetric two-level quantum system. By phase-space analysis, we find that there exist so-called pseudo fixed points in phase space representing the Floquet solutions with fixed population difference and a time-dependent relative phase between the two levels. According to these pseudo fixed points, we analytically construct a non-unitary evolution operator and then explore the dynamic behaviors of the non-PT-symmetric two-level quantum system in different parameter regions. We confirm both analytically and numerically that the two-level non-Hermitian Floquet quantum system, although it is non-parity-time-symmetric, still features a phase transition with the quasienergy spectrum changing from all-real to all-complex energy spectrum, just like the PT symmetric system. Furthermore, we reveal that a novel phenomenon called quasi-PT symmetric dynamics occurs in the time evolution process. The quasi-PT symmetric dynamics is so named in our paper, in the sense that the time-evolution of population probabilities in the non-PT-symmetric two-level system satisfies fully the time-space symmetry (PT symmetry), while time-evolution of the quantum state (containing the phase) does not meet such a PT symmetry, due to the fact that time-evolution of the phases of the probability amplitudes on the two levels violates the requirement for the PT symmetry.
The non-Hermitian models, which are symmetric under parity ( P ) and time-reversal ( T ) operators, are the cornerstone for the fabrication of new ultra-sensitive optoelectronic devices. However, providing the gain in such systems usually demands precise control of nonlinear processes, limiting their application. In this paper, to bypass this obstacle, we introduce a class of time-dependent non-Hermitian Hamiltonians (not necessarily Floquet) that can describe a two-level system with temporally modulated on-site potential and couplings. We show that implementing an appropriate non-Unitary gauge transformation converts the original system to an effective one with a balanced gain and loss. This will allow us to derive the evolution of states analytically. Our proposed class of Hamiltonians can be employed in different platforms such as electronic circuits, acoustics, and photonics to design structures with hidden PT -symmetry potentially without imaginary onsite amplification and absorption mechanism to obtain an exceptional point.
The description of states and dynamics in non-Hermitian systems is fundamentally linked to the choice of an appropriate theoretical framework -- a point of ongoing debate in the field. This work addresses this issue by proposing a consistent formulation that reconciles existing controversies and establishes a unified theoretical understanding. Our approach rests on two foundational premises: (i) the dynamics of both left and right-vectors of a non-Hermitian system must satisfy the Schrödinger equation; (ii) the theoretical framework must preserve gauge invariance, ensuring that physical quantities are independent of unobservable phase choices. Building on these physically motivated assumptions, we refine the biorthogonal framework, leading to a gauge-invariant non-Hermitian quantum theory. Our framework naturally encompasses the open-system effective non-Hermitian evolution as a special case, and can naturally reduce to standard quantum mechanics in the Hermitian limit. As a concrete application, we analyze the dynamical phase transition in a one-dimensional Su-Schrieffer-Heeger (SSH) model within this gauge-invariant non-Hermitian quantum theory. Notably, our formulation naturally generalizes the known condition for such transitions in Hermitian two-band systems, namely, dki⋅dkf=0\mathbf{d}_{k}^i\cdot\mathbf{d}_{k}^f=0dki​⋅dkf​=0, to the non-Hermitian case, where it takes the form Re[dkidki⋅dkfdkf]=0\mathrm{Re}\Bigl[\frac{\mathbf{d}_{k}^i}{d_{k}^i}\cdot\frac{\mathbf{d}_{k}^f}{d_{k}^f}\Bigr]=0Re[dki​dki​​⋅dkf​dkf​​]=0. Furthermore, we identify entirely new dynamical phase transitions that cannot be characterized by the winding number. We hope that this gauge-invariant non-Hermitian quantum theory will find broad applications in the study of non-Hermitian systems.