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.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.

EntropyFeb 28, 20260 citations

Biorthogonal universal quantum control

Chen-yi Zhang, Jun Jing

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 $Λ$ 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.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 $Λ$ 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.

quant-pharXiv:2610.41195Nov 2, 2026

Time, time-dependent perturbation theory, transitions

Gordon Leslie Squires

Cambridge University Press eBooksMar 16, 19950 citations

Effects of non-integrability in a non-Hermitian time crystal

Weihua Xie, Michael Kolodrubetz, Oganesyan, Vadim +1 more

Time crystals are systems that spontaneously break time-translation symmetry, exhibiting repeating patterns in time. Recent work has shown that non-Hermitian Floquet systems can host a time crystalline phase with quasi-long-range order. In this work, we investigate the effect of introducing a non-integrable interaction term into this non-Hermitian time crystal model. Using a combination of numerical TEBD simulations, mean-field analysis, and perturbation theory, we find that the interaction term has two notable effects. First, it induces a shift in the phase diagram, moving the boundaries between different phases. Second, a sufficiently strong interaction induces an unexpected symmetry-breaking transition, which is not captured by the mean-field approach. Within average Hamiltonian theory, we trace this back to a ferromagnetic transition in the anisotropic non-Hermitian XXZ model. Our results demonstrate that the interplay between non-Hermitian dynamics and many-body interactions can lead to novel symmetry breaking.Time crystals are systems that spontaneously break time-translation symmetry, exhibiting repeating patterns in time. Recent work has shown that non-Hermitian Floquet systems can host a time crystalline phase with quasi-long-range order. In this work, we investigate the effect of introducing a non-integrable interaction term into this non-Hermitian time crystal model. Using a combination of numerical TEBD simulations, mean-field analysis, and perturbation theory, we find that the interaction term has two notable effects. First, it induces a shift in the phase diagram, moving the boundaries between different phases. Second, a sufficiently strong interaction induces an unexpected symmetry-breaking transition, which is not captured by the mean-field approach. Within average Hamiltonian theory, we trace this back to a ferromagnetic transition in the anisotropic non-Hermitian XXZ model. Our results demonstrate that the interplay between non-Hermitian dynamics and many-body interactions can lead to novel symmetry breaking.

arXiv:2412.04382Dec 5, 20240 citations

Time-Dependent Perturbation Theory

Stephen P. Martin

Graduate texts in physicsJan 1, 20260 citations

Λ

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.