We investigate the geometric phases and the Bargmann invariants associated with multilevel quantum systems. In particular, we show that a full set of ``gauge-invariant'' objects for an n-level system consists of n geometric phases and $\frac{1}{2}(n\ensuremath{-}1)(n\ensuremath{-}2)$ algebraically independent four-vertex Bargmann invariants. In the process of establishing this result, we develop a canonical form for $\mathrm{U}(n)$ matrices that is useful in its own right. We show that the recently discovered ``off-diagonal'' geometric phases [N. Manini and F. Pistolesi, Phys. Rev. Lett. 8, 3067 (2000)] can be completely analyzed in terms of the basic building blocks developed in this work. This result liberates the off-diagonal phases from the assumption of adiabaticity used in arriving at them.We investigate the geometric phases and the Bargmann invariants associated with multilevel quantum systems. In particular, we show that a full set of ``gauge-invariant'' objects for an n-level system consists of n geometric phases and $\frac{1}{2}(n\ensuremath{-}1)(n\ensuremath{-}2)$ algebraically independent four-vertex Bargmann invariants. In the process of establishing this result, we develop a canonical form for $\mathrm{U}(n)$ matrices that is useful in its own right. We show that the recently discovered ``off-diagonal'' geometric phases [N. Manini and F. Pistolesi, Phys. Rev. Lett. 8, 3067 (2000)] can be completely analyzed in terms of the basic building blocks developed in this work. This result liberates the off-diagonal phases from the assumption of adiabaticity used in arriving at them.

arXiv:quant-ph/0107006Physical Review ADec 10, 200136 citations

New derivation of the geometric phase

Arun Kumar Pati

Physics Letters AJun 1, 199534 citations

Dynamical invariants and nonadiabatic geometric phases in open quantum systems

M. S. Sarandy, Eduardo I. Duzzioni, M. H. Y. Moussa

We introduce an operational framework to analyze nonadiabatic Abelian and non-Abelian, cyclic and noncyclic, geometric phases in open quantum systems. In order to remove the adiabaticity condition, we generalize the theory of dynamical invariants to the context of open systems evolving under arbitrary convolutionless master equations. Geometric phases are then defined through the Jordan canonical form of the dynamical invariant associated with the superoperator that governs the master equation. As a by-product, we provide a sufficient condition for the robustness of the phase against a given decohering process. We illustrate our results by considering a two-level system in a Markovian interaction with the environment, where we show that the nonadiabatic geometric phase acquired by the system can be constructed in such a way that it is robust against both dephasing and spontaneous emission.We introduce an operational framework to analyze nonadiabatic Abelian and non-Abelian, cyclic and noncyclic, geometric phases in open quantum systems. In order to remove the adiabaticity condition, we generalize the theory of dynamical invariants to the context of open systems evolving under arbitrary convolutionless master equations. Geometric phases are then defined through the Jordan canonical form of the dynamical invariant associated with the superoperator that governs the master equation. As a by-product, we provide a sufficient condition for the robustness of the phase against a given decohering process. We illustrate our results by considering a two-level system in a Markovian interaction with the environment, where we show that the nonadiabatic geometric phase acquired by the system can be constructed in such a way that it is robust against both dephasing and spontaneous emission.

arXiv:0707.2579Physical Review ANov 20, 200728 citations

Geometric phases and related structures

Armin Uhlmann

Reports on Mathematical PhysicsOct 1, 199594 citations

A geometrical quantum phase effect

Jean-Marc Lévy-Leblond

Physics Letters ADec 1, 198736 citations

Emergence of Lie group symmetric classical spacetimes in the canonical tensor model

Taigen Kawano, Naoki Sasakura

Abstract We analyze a wave function of a tensor model in the canonical formalism, when the argument of the wave function takes Lie group invariant or nearby values. Numerical computations show that there are two phases, which we call the quantum and the classical phases. In the classical phase fluctuations are suppressed, and configurations emerge which are discretizations of classical geometric spaces invariant under Lie group symmetries. This is explicitly demonstrated for emergence of Sn (n = 1, 2, 3) for SO(n + 1) symmetries by checking the topological and geometric (Laplacian) properties of the emerging configurations. The transition between the two phases has the form of splitting/merging of distributions of variables, resembling a matrix model counterpart, namely the transition between one-cut and two-cut solutions. However, this resemblance is obscured by a difference in the mechanism of distribution in our setup from that in the matrix model. We also discuss this transition as a replica symmetry breaking. We perform various preliminary studies of the properties of the phases and the transition for such values of the argument.Abstract We analyze a wave function of a tensor model in the canonical formalism, when the argument of the wave function takes Lie group invariant or nearby values. Numerical computations show that there are two phases, which we call the quantum and the classical phases. In the classical phase fluctuations are suppressed, and configurations emerge which are discretizations of classical geometric spaces invariant under Lie group symmetries. This is explicitly demonstrated for emergence of Sn (n = 1, 2, 3) for SO(n + 1) symmetries by checking the topological and geometric (Laplacian) properties of the emerging configurations. The transition between the two phases has the form of splitting/merging of distributions of variables, resembling a matrix model counterpart, namely the transition between one-cut and two-cut solutions. However, this resemblance is obscured by a difference in the mechanism of distribution in our setup from that in the matrix model. We also discuss this transition as a replica symmetry breaking. We perform various preliminary studies of the properties of the phases and the transition for such values of the argument.

arXiv:2109.09896Progress of Theoretical and Experimental PhysicsMar 7, 20228 citations

\mathrm{U}(n)

matrices that is useful in its own right. We show that the recently discovered ``off-diagonal'' geometric phases [N. Manini and F. Pistolesi, Phys. Rev. Lett. 8, 3067 (2000)] can be completely analyzed in terms of the basic building blocks developed in this work. This result liberates the off-diagonal phases from the assumption of adiabaticity used in arriving at them.