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Geometric phases for three level nonHermitian system

S.E. Korenblit, V. A. Naumov, Valentin E. Kuznetsov

Published Jan 1, 1994• 0 citations
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Abstract

The complex geometric phases for systems governed by non-Hermitian 3×3 matrix Hamiltonians depending on time through a set of real parameters are constructed. The case that the non-Hermitian three-level system undergoes a cyclic evolution is considered for the class of the Hamiltonians with time-independent off-diagonal matrix elements. The conditions are established whereby the complex geometric phases take their extreme (resonance) values independently of the circuit in the parameter space. The results can be in particular used to study the problem of three-neutrino oscillations with regard to the neutrino refraction and absorption in matter.

Related papers

1 A New Class of Adiabatic Cyclic States and Geometric Phases for Non-Hermitian Hamiltonians

For a T-periodic non-Hermitian Hamiltonian H(t), we construct a class of adiabatic cyclic states of period T which are not eigenstates of the initial Hamiltonian H(0). We show that the corresponding adiabatic geometric phase angles are real and discuss their relationship with the conventional complex adiabatic geometric phase angles. We present a detailed calculation of the new adiabatic cyclic states and their geometric phases for a non-Hermitian analog of the spin 1/2 particle in a precessing magnetic field. Since the publication of Berry’s paper [1] on the adiabatic geometrical phase, the subject has undergone a rapid development. Berry’s adiabatic geometric phase for periodic Hermitian Hamiltonians with a discrete nondegenerate spectrum has been generalized to arbitrary changes of a quantum state. In particular, the conditions on the adiabaticity [2] and cyclicity [3, 4, 5] of the evolution, Hermiticity of the Hamiltonian [6], and degeneracy [7] and discreteness [8] of its spectrum have been lifted. Moreover, the classical [9] and relativistic [10] analogues of the geometric phase have been considered. The purpose of this note is to offer an alternative generalization of Berry’s treatment of the adiabatic geometric phase for a non-Hermitian parametric Hamiltonian H[R] with a nondegenerate discrete spectrum. The parameters R = (R1, · · ·, Rn) are assumed to be real ∗E-mail address:

•Jan 1, 1999•0 citations

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In this paper, the Lewis-Riesenfeld invariant theory is generalized for the study of systems with non-Hermitian time-dependent Hamiltonians. It is then used to study the nonadiabatic cyclic evolution and the Aharonov-Anandan phase. It is shown that the study of noncyclic evolution can be reduced to the study of cyclic evolution. The two-level dissipative system and the classical time-dependent harmonic oscillator are discussed as illustrative examples.

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The geometric picture of neutrino oscillations offers a unique way to study the quantum mechanics of this phenomenon. In this picture, the propagation of a neutrino beam is described by a density matrix evolving in a state space with non-trivial geometry. We derive explicit expressions of the mixed state geometric phase which arise during such an evolution for both two and three flavor neutrino oscillations. We show that, in the case of two flavor neutrino oscillations, the geometric phase is independent of the Majorana phase and it can be used as a measure of coherence of the neutrino beam.

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Geometric phases in non-Hermitian quantum mechanics

Xiaodong Cui, Yujun Zheng

We generally derive complex geometric phases in non-Hermitian quantum mechanics. The formalism of complex geometric phases for non-Hermitian quantum systems is general and can be applied in the cases of cyclic and noncyclic states in non-Hermitian quantum mechanics. As an example, we show the interpretation of complex geometric phases by a generalized density matrix defined on a two-level non-Hermitian quantum system and the two-level Bethe-Lamb model. Also, in the Hermitian case, our theoretical framework of a complex geometric phase reduces to the counterpart of the Pancharatnam phase.

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Geometric modulation of transition and survival intensities in non-Hermitian systems

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The time evolution of non-Hermitian systems is generally nonunitary. Dynamics governed by time-dependent non-Hermitian Hamiltonians lead to a variety of novel phenomena, one of which is state amplification or suppression induced by the complex Berry phase. Here, we extend the framework of geometric modulation to multi-level systems and show that both transition and survival intensities can be modulated. We apply our theory to the non-Hermitian Landau-Zener (LZ) problem. First, we show that, in the half-LZ problem, both the transition and survival probabilities exhibit nonreciprocity due to the complex Berry phase. In the non-Hermitian standard LZ problem, only the survival intensity is known to exhibit nonreciprocity, whereas the transition intensity does not. However, the physical origin of this nonreciprocal behavior remains unclear. In this work, we show that the nonreciprocity originates from the complex Berry phase.

quant-pharXiv:2607.03164•Physical Review A•Jul 6, 2026•0 citations

Periodic pseudo-Hermitian Hamiltonian: Nonadiabatic geometric phase

Mustapha Maamache

It is well known that Hermitian operators have real eigenvalues while non-Hermitian ones may have complex eigenvalues. Recently, numerical and analytical results indicated that the spectra of many non-Hermitians Hamiltonians $H$ are indeed real if they are invariant under the combined action of self-adjoint parity $P$ and time reversal $T.$ The concept of a pseudo-Hermitian operator showed that the remarkable spectral properties of the $PT$-symmetric Hamiltonians follow from their pseudo-Hermiticity. It is possible to explain these observations by the concept of pseudo-Hermitian operators and to formulate completeness and orthonormality relations. Most of the effort has been devoted to study time-independent non-Hermitian systems. In this paper, we study the exactly solvable time-dependent periodic pseudo-Hermitian Hamiltonians. The method introduced, to make the reality of eigenvalues and phases, is based on a Floquet decomposition of the evolution operator ${U}^{H}(t)={Z}^{H}(t)exp(i{M}^{H}t)$ associated with the periodic pseudo-hermitian Hamiltonian $H(t)=H(t+T)$. One of the results found in this paper concerns a calculation of Berry's phase for periodic, but not necessarily adiabatic, pseudo-Hermitian Hamiltonians. A two-level pseudo-Hermitian system is discussed as an illustrative example.

Physical Review A•Sep 8, 2015•29 citations
The complex geometric phases for systems governed by non-Hermitian 3×3 matrix Hamiltonians depending on time through a set of real parameters are constructed. The case that the non-Hermitian three-level system undergoes a cyclic evolution is considered for the class of the Hamiltonians with time-independent off-diagonal matrix elements. The conditions are established whereby the complex geometric phases take their extreme (resonance) values independently of the circuit in the parameter space. The results can be in particular used to study the problem of three-neutrino oscillations with regard to the neutrino refraction and absorption in matter.
For a T-periodic non-Hermitian Hamiltonian H(t), we construct a class of adiabatic cyclic states of period T which are not eigenstates of the initial Hamiltonian H(0). We show that the corresponding adiabatic geometric phase angles are real and discuss their relationship with the conventional complex adiabatic geometric phase angles. We present a detailed calculation of the new adiabatic cyclic states and their geometric phases for a non-Hermitian analog of the spin 1/2 particle in a precessing magnetic field. Since the publication of Berry’s paper [1] on the adiabatic geometrical phase, the subject has undergone a rapid development. Berry’s adiabatic geometric phase for periodic Hermitian Hamiltonians with a discrete nondegenerate spectrum has been generalized to arbitrary changes of a quantum state. In particular, the conditions on the adiabaticity [2] and cyclicity [3, 4, 5] of the evolution, Hermiticity of the Hamiltonian [6], and degeneracy [7] and discreteness [8] of its spectrum have been lifted. Moreover, the classical [9] and relativistic [10] analogues of the geometric phase have been considered. The purpose of this note is to offer an alternative generalization of Berry’s treatment of the adiabatic geometric phase for a non-Hermitian parametric Hamiltonian H[R] with a nondegenerate discrete spectrum. The parameters R = (R1, · · ·, Rn) are assumed to be real ∗E-mail address:
The geometric picture of neutrino oscillations offers a unique way to study the quantum mechanics of this phenomenon. In this picture, the propagation of a neutrino beam is described by a density matrix evolving in a state space with non-trivial geometry. We derive explicit expressions of the mixed state geometric phase which arise during such an evolution for both two and three flavor neutrino oscillations. We show that, in the case of two flavor neutrino oscillations, the geometric phase is independent of the Majorana phase and it can be used as a measure of coherence of the neutrino beam.
We generally derive complex geometric phases in non-Hermitian quantum mechanics. The formalism of complex geometric phases for non-Hermitian quantum systems is general and can be applied in the cases of cyclic and noncyclic states in non-Hermitian quantum mechanics. As an example, we show the interpretation of complex geometric phases by a generalized density matrix defined on a two-level non-Hermitian quantum system and the two-level Bethe-Lamb model. Also, in the Hermitian case, our theoretical framework of a complex geometric phase reduces to the counterpart of the Pancharatnam phase.
The time evolution of non-Hermitian systems is generally nonunitary. Dynamics governed by time-dependent non-Hermitian Hamiltonians lead to a variety of novel phenomena, one of which is state amplification or suppression induced by the complex Berry phase. Here, we extend the framework of geometric modulation to multi-level systems and show that both transition and survival intensities can be modulated. We apply our theory to the non-Hermitian Landau-Zener (LZ) problem. First, we show that, in the half-LZ problem, both the transition and survival probabilities exhibit nonreciprocity due to the complex Berry phase. In the non-Hermitian standard LZ problem, only the survival intensity is known to exhibit nonreciprocity, whereas the transition intensity does not. However, the physical origin of this nonreciprocal behavior remains unclear. In this work, we show that the nonreciprocity originates from the complex Berry phase.
It is well known that Hermitian operators have real eigenvalues while non-Hermitian ones may have complex eigenvalues. Recently, numerical and analytical results indicated that the spectra of many non-Hermitians Hamiltonians HHH are indeed real if they are invariant under the combined action of self-adjoint parity PPP and time reversal T.T.T. The concept of a pseudo-Hermitian operator showed that the remarkable spectral properties of the PTPTPT-symmetric Hamiltonians follow from their pseudo-Hermiticity. It is possible to explain these observations by the concept of pseudo-Hermitian operators and to formulate completeness and orthonormality relations. Most of the effort has been devoted to study time-independent non-Hermitian systems. In this paper, we study the exactly solvable time-dependent periodic pseudo-Hermitian Hamiltonians. The method introduced, to make the reality of eigenvalues and phases, is based on a Floquet decomposition of the evolution operator UH(t)=ZH(t)exp(iMHt){U}^{H}(t)={Z}^{H}(t)exp(i{M}^{H}t)UH(t)=ZH(t)exp(iMHt) associated with the periodic pseudo-hermitian Hamiltonian H(t)=H(t+T)H(t)=H(t+T)H(t)=H(t+T). One of the results found in this paper concerns a calculation of Berry's phase for periodic, but not necessarily adiabatic, pseudo-Hermitian Hamiltonians. A two-level pseudo-Hermitian system is discussed as an illustrative example.