Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:0706.0320

Properties of Geometric Potential in the Invariant Adiabatic Theory

Meisheng Zhao, Jianda Wu, Jian-lan Chen, Zhang, Yong-de

Published Jun 3, 2007• 1 citation
arXivHTMLPDF

Abstract

We concentrate on the geometric potential in the invariant perturbation theory of quantum adiabatic process which is presented in our recent papers. It is found out to be related to the geodesic curvature of the spherical curve in 2-dimension quantum systems. We also show that the geometric potential may affect adiabatic approximation remarkably.

DOI
10.48550/arxiv.0706.0320

Related papers

Adiabatic condition and quantum geometric potential

Jianda Wu, Meisheng Zhao, Jian-lan Chen +1 more

In this paper, we present a $\text{U}(1)$-invariant expansion theory of the adiabatic process. As its application, we propose and discuss different sufficient adiabatic approximation conditions. In these conditions, we find an interesting invariant quantity referred to as the quantum geometric potential (QGP) contained in all time-dependent processes. Furthermore, we also give detailed discussion and analysis on the properties and effects of QGP.

arXiv:0803.3359•Physical Review A•Jun 25, 2008•33 citations

Quantum adiabatic approximation condition and quantum geometric potential

YI Xue-hu

The quantum traditional adiabatic theorem was present and traditional approximation conditions were given.Based on existence of insufficence about quntum traditional adiabatic approximation condition,the new adiabatic theory and approximation condition were proposed and discussed by means of perturbative approach and U(1)-invariant expasion.The properties of quantum geometric potential contained in the new adiabatic condition were analyzed and discussed.

Chinese Journal of Quantum Electronics•Jan 1, 2014•0 citations

Invariant Perturbation Theory of Adiabatic Process

Jianlan Chen, Meisheng Zhao, Jianda Wu +1 more

In this paper we present an invariant perturbation theory to adiabatic process according to the concepts of adiabatic orbits, adiabatic evolution orbit and U(1)-invariant adiabatic orbit. The probabilities of keeping the adiabatic orbit in the first-order and the second-order approximation are calculated, respectively. We also give a convenient sufficient condition.

arXiv:0706.0299•Jun 3, 2007•2 citations

Quantum geometry of the parameter space: a proposal for curved systems

Joshua Davy-Castillo, Javier Alexander Cano-Arango, Sergio B. Juárez +2 more

In this paper, we extend the quantum geometric tensor for parameter-dependent curved spaces to higher dimensions, and introduce an equivalent definition that generalizes the Zanardi, et al, formulation of the tensor. The parameter-dependent metric modifies the behavior of both the quantum metric tensor and Berry curvature in a purely geometric way. Our focus is on understanding the distinctions in higher dimensions that emerge when using the generalized tensor compared to the conventional one. Through a comparative analysis, illustrated with examples in two dimensions, we highlight unique quantum geometric properties for both the quantum metric tensor and the Berry curvature. Additionally, we explore differences between analytical and perturbative approaches in solving the problems.

arXiv:2403.09804•Mar 14, 2024•0 citations

A proposal for direct measurement on the quantum geometric potential

Jianda Wu

The quantum geometric potential is a gauge invariant carrying novel geometric features between any two energy levels or bands in quantum systems. In generic time-dependent systems it gives a vital physical modification for the instantaneous energy gaps, laying down more appropriate quantum adiabatic conditions for both non-degenerate and degenerate systems. Remarkably, for generic parameterized quantum systems, the integration of the quantum geometric potential on a closed loop leads to a novel type of quantized winding number, which is a quantum counterpart of the Gauss-Bonnet theorem. The effects of the quantum geometric potential had been indirectly supported by the experiments on the quantum adiabatic evolution, however, a direct experimental observation so far is lacking. In this paper we propose an interference measurement to directly probe the quantum geometric potential, where the relevant parameters are easily accessible by current experimental apparatus. A direct confirmation of this new physical quantity could motive further theoretical and experimental investigations as well as its potential real applications.

arXiv:1801.00190•Dec 30, 2017•0 citations

Adiabatic Approximation for Path Integrals and Geometrical Potentials

Takashi Ohnishi, Naoto Nagaosa

Geometrical concepts have played an important role in quantum mechanics. In the Schrodinger representation, the Berry–Simon connection and the quantum geometric tensor appear as gauge potentials in the effective Hamiltonian for a heavy system under the Born–Oppenheimer approximation. On the other hand, the Berry phase should arise as the amplitude of each path in the pathintegral formalism. In their pioneering work, Kuratsuji and Iida have developed the treatment of the Born– Oppenheimer approximation in the path integral quantization, where the Berry phase of the light system naturally appears in the effective action for the heavy system. However, they did not derive the gauge potentials themselves in the effective Hamiltonian. In this short note, we show that the gauge potentials which relate to the geometry of the Hilbert space of the light system are also derived in the adiabatic scheme of path integrals. Our result presents the path-integral formalism to consider the effective dynamics of the heavy system under the adiabatic approximation. Let us consider the following Hamiltonian:

Journal of the Physical Society of Japan•Dec 25, 2006•2 citations
We concentrate on the geometric potential in the invariant perturbation theory of quantum adiabatic process which is presented in our recent papers. It is found out to be related to the geodesic curvature of the spherical curve in 2-dimension quantum systems. We also show that the geometric potential may affect adiabatic approximation remarkably.
In this paper, we present a U(1)\text{U}(1)U(1)-invariant expansion theory of the adiabatic process. As its application, we propose and discuss different sufficient adiabatic approximation conditions. In these conditions, we find an interesting invariant quantity referred to as the quantum geometric potential (QGP) contained in all time-dependent processes. Furthermore, we also give detailed discussion and analysis on the properties and effects of QGP.
In this paper, we extend the quantum geometric tensor for parameter-dependent curved spaces to higher dimensions, and introduce an equivalent definition that generalizes the Zanardi, et al, formulation of the tensor. The parameter-dependent metric modifies the behavior of both the quantum metric tensor and Berry curvature in a purely geometric way. Our focus is on understanding the distinctions in higher dimensions that emerge when using the generalized tensor compared to the conventional one. Through a comparative analysis, illustrated with examples in two dimensions, we highlight unique quantum geometric properties for both the quantum metric tensor and the Berry curvature. Additionally, we explore differences between analytical and perturbative approaches in solving the problems.
The quantum geometric potential is a gauge invariant carrying novel geometric features between any two energy levels or bands in quantum systems. In generic time-dependent systems it gives a vital physical modification for the instantaneous energy gaps, laying down more appropriate quantum adiabatic conditions for both non-degenerate and degenerate systems. Remarkably, for generic parameterized quantum systems, the integration of the quantum geometric potential on a closed loop leads to a novel type of quantized winding number, which is a quantum counterpart of the Gauss-Bonnet theorem. The effects of the quantum geometric potential had been indirectly supported by the experiments on the quantum adiabatic evolution, however, a direct experimental observation so far is lacking. In this paper we propose an interference measurement to directly probe the quantum geometric potential, where the relevant parameters are easily accessible by current experimental apparatus. A direct confirmation of this new physical quantity could motive further theoretical and experimental investigations as well as its potential real applications.
Geometrical concepts have played an important role in quantum mechanics. In the Schrodinger representation, the Berry–Simon connection and the quantum geometric tensor appear as gauge potentials in the effective Hamiltonian for a heavy system under the Born–Oppenheimer approximation. On the other hand, the Berry phase should arise as the amplitude of each path in the pathintegral formalism. In their pioneering work, Kuratsuji and Iida have developed the treatment of the Born– Oppenheimer approximation in the path integral quantization, where the Berry phase of the light system naturally appears in the effective action for the heavy system. However, they did not derive the gauge potentials themselves in the effective Hamiltonian. In this short note, we show that the gauge potentials which relate to the geometry of the Hilbert space of the light system are also derived in the adiabatic scheme of path integrals. Our result presents the path-integral formalism to consider the effective dynamics of the heavy system under the adiabatic approximation. Let us consider the following Hamiltonian: