© 2026 Aqora Quantum S.A.S.

Terms Privacy Legal Notice

Physica Scripta

Explicit form of Berry phase for time dependent harmonic oscillators in noncommutative space

Manjari Dutta, Shreemoyee Ganguly, Sunandan Gangopadhyay

Published Aug 30, 2022• 4 citations

Abstract

Abstract In this paper we look for the existence of Berry phase in time dependent harmonic oscillators in noncommutative space. Two systems are considered in our study in the noncommutative framework. The first one is a system in which a scale invariant term is present in the Hamiltonian from the beginning, and the second one is a system in which a scale invariant term emerges due to a change of variables from the noncommutative to the commutative ones. We first compute the eigenstates of both the systems using the Lewis invariant approach. We then employ the Lewis invariant technique to obtain the geometric phase under adiabatic approximation. We also calculate the Berry phase explicitly by choosing appropriate forms of the time dependent parameters appearing in the Hamiltonian. Our analysis surprisingly reveals that a scale invariant time reversal symmetry breaking term may not always lead to a non trivial Berry phase.

DOI: 10.1088/1402-4896/ac8dca

Related papers

Berry Phase for Time-Dependent Coupled Harmonic Oscillators in the Noncommutative Phase Space via Path Integral Techniques

Leila Khiari, Tahar Boudjedaa, Abdenacer Makhlouf +1 more

The purpose of this paper is the description of Berry’s phase, in the Euclidean Path Integral formalism, for 2D quadratic system: two time dependent coupled harmonic oscillators. This treatment is achieved by using the adiabatic approximation in the commutative and noncommutative phase space

Journal of Siberian Federal University Mathematics & Physics•Feb 1, 2020•0 citations

Emergence of a geometric phase shift in planar noncommutative quantum mechanics

Saptarshi Biswas, Partha Nandi, Biswajit Chakraborty

Appearance of adiabatic geometric phase shift in the context of noncommutative quantum mechanics is studied using an exactly solvable model of a two-dimensional simple harmonic oscillator in the Moyal plane, where momentum noncommutativity are also considered along with spatial noncommutativity. For that we introduce a modified form of Bopp's shift, that bridges the noncommutative phase-space operators with their effective commutative counterparts, having their dependence on the noncommutative parameters, and study the adiabatic evolution in the Heisenberg picture. An explicit expression for the geometric phase shift under adiabatic approximation is then found without using any perturbative technique. Lastly, this phase is found to be related to Hannay's angle of a classically analogous system, by studying the evolution of the coherent state of this system.

arXiv:1911.03196•Physical Review A•Aug 28, 2020•20 citations

Behavioral Differences of a Time-Dependent Harmonic Oscillator in Commutative Space and Non-Commutative Phase Space

Hadi Sobhani, HASSAN HASSANABADI

In this article, behavioral differences of time-dependent harmonic oscillator in Commutative space and Non-Commutative phase space have been investigated. The considered harmonic oscillator has a time-dependent angular frequency and mass which are function of time. First, the time-dependent harmonic oscillator is studied in commutative space, then similar calculation is done for considered harmonic oscillator in Non-Commutative phase space. During this article method of Lewis–Riesenfeld dynamical invariant has been employed.

Physics of Particles and Nuclei Letters•Sep 1, 2018•2 citations

Invariance of the generalised Berry phase under unitary transformations: application to the time-dependent generalised harmonic oscillator

Donald H. Kobe

A generalisation of the Berry phase, which reduces to the usual Berry phase in the adiabatic and cyclic case is derived in a gauge-invariant way. This generalised Berry phase, unlike the Berry phase, is invariant under unitary transformations. A time-dependent generalised harmonic oscillator with a cross term in the Hamiltonian is taken as an example. The Hamiltonian of this system is not in general the energy. An energy, the time derivative of which is the power, is obtained from the equation of motion. When the system is quantised, the generalised Berry phase is zero, and remains invariant under unitary transformations. On the other hand, if the energy is chosen to be the Hamiltonian, a non-zero generalised Berry phase is obtained, but the power is given incorrectly. In this case the total phase, which includes the dynamical phase and the generalised Berry phase, is equal in the adiabatic limit to the correct total phase.

Journal of Physics A Mathematical and General•Oct 7, 1990•26 citations

Some Studies On Exact Solutions Of Models In Noncommutative Spaces

Manjari Dutta

The central theme of my thesis is to explore various simple prototype models that are exactly solvable in the framework of time dependent noncommutative spaces. By adopting the methodology provided by the Lewis Riesenfeld theory, we developed a procedure for obtaining a class of exact solutions for such model systems. We analyzed these solutions by deriving the energy expectation values analytically and representing those energy dynamics graphically. We also examined the explicit existence of a non-zero Berry geometric phase in the noncommutative framework and analyzed the role of noncommutativity in generating a non-trivial Berry phase when the model Hamiltonian and the noncommutative parameters are periodic in time. Overall, my thesis contributes to a deeper understanding of quantum theory in time dependent noncommutative backgrounds and indicates a strong possibility for developing a consistent quantum theory within such frameworks.

quant-pharXiv:2603.18047•Mar 18, 2026•0 citations

Fingerprints of the quantum space-time in time-dependent quantum mechanics: An emergent geometric phase

Anwesha Chakraborty, Partha Nandi, Biswajit Chakraborty

We show the emergence of Berry phase in a forced harmonic oscillator system placed in the quantum space-time of Moyal type, where the time ‘t’ is also an operator. An effective commutative description of the system gives a time dependent generalised harmonic oscillator system with perturbation linear in position and momentum. The system is then diagonalised to get a generalised harmonic oscillator and then its adiabatic evolution over time-period T is studied in Heisenberg picture to compute the expression of geometric phase-shift.

arXiv:2110.04730•Nuclear Physics B•Jan 29, 2022•8 citations

Abstract In this paper we look for the existence of Berry phase in time dependent harmonic oscillators in noncommutative space. Two systems are considered in our study in the noncommutative framework. The first one is a system in which a scale invariant term is present in the Hamiltonian from the beginning, and the second one is a system in which a scale invariant term emerges due to a change of variables from the noncommutative to the commutative ones. We first compute the eigenstates of both the systems using the Lewis invariant approach. We then employ the Lewis invariant technique to obtain the geometric phase under adiabatic approximation. We also calculate the Berry phase explicitly by choosing appropriate forms of the time dependent parameters appearing in the Hamiltonian. Our analysis surprisingly reveals that a scale invariant time reversal symmetry breaking term may not always lead to a non trivial Berry phase.

Appearance of adiabatic geometric phase shift in the context of noncommutative quantum mechanics is studied using an exactly solvable model of a two-dimensional simple harmonic oscillator in the Moyal plane, where momentum noncommutativity are also considered along with spatial noncommutativity. For that we introduce a modified form of Bopp's shift, that bridges the noncommutative phase-space operators with their effective commutative counterparts, having their dependence on the noncommutative parameters, and study the adiabatic evolution in the Heisenberg picture. An explicit expression for the geometric phase shift under adiabatic approximation is then found without using any perturbative technique. Lastly, this phase is found to be related to Hannay's angle of a classically analogous system, by studying the evolution of the coherent state of this system.

In this article, behavioral differences of time-dependent harmonic oscillator in Commutative space and Non-Commutative phase space have been investigated. The considered harmonic oscillator has a time-dependent angular frequency and mass which are function of time. First, the time-dependent harmonic oscillator is studied in commutative space, then similar calculation is done for considered harmonic oscillator in Non-Commutative phase space. During this article method of Lewis–Riesenfeld dynamical invariant has been employed.

A generalisation of the Berry phase, which reduces to the usual Berry phase in the adiabatic and cyclic case is derived in a gauge-invariant way. This generalised Berry phase, unlike the Berry phase, is invariant under unitary transformations. A time-dependent generalised harmonic oscillator with a cross term in the Hamiltonian is taken as an example. The Hamiltonian of this system is not in general the energy. An energy, the time derivative of which is the power, is obtained from the equation of motion. When the system is quantised, the generalised Berry phase is zero, and remains invariant under unitary transformations. On the other hand, if the energy is chosen to be the Hamiltonian, a non-zero generalised Berry phase is obtained, but the power is given incorrectly. In this case the total phase, which includes the dynamical phase and the generalised Berry phase, is equal in the adiabatic limit to the correct total phase.

The central theme of my thesis is to explore various simple prototype models that are exactly solvable in the framework of time dependent noncommutative spaces. By adopting the methodology provided by the Lewis Riesenfeld theory, we developed a procedure for obtaining a class of exact solutions for such model systems. We analyzed these solutions by deriving the energy expectation values analytically and representing those energy dynamics graphically. We also examined the explicit existence of a non-zero Berry geometric phase in the noncommutative framework and analyzed the role of noncommutativity in generating a non-trivial Berry phase when the model Hamiltonian and the noncommutative parameters are periodic in time. Overall, my thesis contributes to a deeper understanding of quantum theory in time dependent noncommutative backgrounds and indicates a strong possibility for developing a consistent quantum theory within such frameworks.

We show the emergence of Berry phase in a forced harmonic oscillator system placed in the quantum space-time of Moyal type, where the time ‘t’ is also an operator. An effective commutative description of the system gives a time dependent generalised harmonic oscillator system with perturbation linear in position and momentum. The system is then diagonalised to get a generalised harmonic oscillator and then its adiabatic evolution over time-period T is studied in Heisenberg picture to compute the expression of geometric phase-shift.