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arXiv:2206.06748 · Journal of Physics A Mathematical and Theoretical

Consistency between adiabatic and non-adiabatic geometric phases for non-self-adjoint Hamiltonians

David Viennot, Arnaud Leclerc, Georges Jolicard, John P. Killingbeck

Published Jul 26, 2012• 6 citations
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Abstract

We show that the adiabatic approximation for non-self-adjoint Hamiltonians seems to induce two non-equal expressions for the geometric phase. The first one is related to the spectral projector involved in the adiabatic theorem, and the other one is the adiabatic limit of the non-adiabatic geometric phase. This apparent inconsistency is resolved by observing that the difference between the two expressions is compensated by a small deviation in the dynamical phases.

DOI
10.1088/1751-8113/45/33/335301

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We show that the adiabatic approximation for non-self-adjoint Hamiltonians seems to induce two non-equal expressions for the geometric phase. The first one is related to the spectral projector involved in the adiabatic theorem, and the other one is the adiabatic limit of the non-adiabatic geometric phase. This apparent inconsistency is resolved by observing that the difference between the two expressions is compensated by a small deviation in the dynamical phases.
We show that the definition of instantaneous eigenstate populations for a dynamical non-self-adjoint system is not obvious. The naive direct extension of the definition used for the self-adjoint case leads to inconsistencies; the resulting artifacts can induce a false inversion of population or a false adiabaticity. We show that the inconsistency can be avoided by introducing geometric phases in another possible definition of populations. An example is given which demonstrates both the anomalous effects and their removal by our approach.
We unveil that even an infinitesimal deviation of the initial state from the eigenstates of the initial Hamiltonian can yield a significant change in the geometric phase accompanying adiabatic evolution without degeneracies of the energy levels. This leads to the counterintuitive result that the values of two geometric phases computed from the general definition of the Berry phase [Samuel and Bhandari, Phys. Rev. Lett. 60, 2339 (1988)] can dramatically differ from each other in the adiabatic limit even if their evolution paths are sufficiently close in a pointwise manner. The result of this work indicates that the widely accepted theory of geometric phases is not very consistent in the context of adiabatic evolution involving superposed eigenstates.
We show that the definition of instantaneous eigenstate populations for a dynamical non-selfadjoint system is not obvious. The naive direct extension of the definition used for the self-adjoint case leads to inconsistencies; the resulting artifacts can induce a false inversion of population or a false adiabaticity. We show that the inconsistency can be avoided by introducing geometric phases in another possible definition of populations. An example is given which demonstrates both the anomalous effects and their removal by our approach.
The authors consider the evolution of a two-level system driven by a nonself-adjoint Hamiltonian H( in t) and treat the adiabatic limit in to 0. While adiabatic theorem-like results do not hold true in general for this case, they prove that they are still valid for the subspace corresponding to the eigenvalue having the largest imaginary part (least dissipative eigenvalue). The theory gives the full asymptotic expansion of the evolution restricted to this subspace. The first correction beyond Berry's phase is to their best knowledge given explicitly for the first time.
We construct the adiabatic theorem independent of the quantum representations. The quantum-representation-independent geometrical phase, adiabatic condition, and fidelity of the adiabatic approximation are derived. As an example, we apply the quantum-representation-independent adiabatic theorem to provide a thorough resolution of the Marzlin-Sanders inconsistency. We show that the Marzlin-Sanders inconsistency results from the wrong identification of the time-dependent Hamiltonian as the instantaneous energy operator and the quantum representation dependence of the Marzlin-Sanders transformation.