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arXiv:2307.00901 · Physica Scripta

Lewis and berry phases for a gravitational wave interacting with a quantum harmonic oscillator

Soham Sen, Manjari Dutta, Sunandan Gangopadhyay

Published Dec 4, 2023• 4 citations

Abstract

Abstract In this work, we compute the Lewis and Berry phases for a gravitational wave interacting with a two dimensional quantum harmonic oscillator in the transverse-traceless gauge. We have considered a gravitational wave consisting of the plus polarization term only. Considering the cross polarization term to be absent makes the Hamiltonian separable in terms of the first and the second spatial coordinates. We then compute the Lewis phase by assuming a suitable form of the Lewis invariant considering only quadratic order contributions from both position and momentum variables. Next, we obtain two Lewis invariants corresponding to each separable part of the full Hamiltonian of the system. Using both Lewis invariants, one can obtain two Ermakov-Pinney equations, from which we finally obtain the corresponding Lewis phase. Then making an adiabatic approximation enables us to isolate the Berry phase for the full system. After this we obtain some explicit expressions of the Berry phase for a plane polarized gravitational wave with different choices of the harmonic oscillator frequency. Finally, we consider a gravitational wave with cross polarization only interacting with an isotropic two dimensional harmonic oscillator. For this we obtain the Lewis phase and the total Berry phase of the system, which is found to be dependent upon the cross polarization part of the gravitational wave.

DOI: 10.1088/1402-4896/ad1234

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Abstract In this work, we compute the Lewis and Berry phases for a gravitational wave interacting with a two dimensional quantum harmonic oscillator in the transverse-traceless gauge. We have considered a gravitational wave consisting of the plus polarization term only. Considering the cross polarization term to be absent makes the Hamiltonian separable in terms of the first and the second spatial coordinates. We then compute the Lewis phase by assuming a suitable form of the Lewis invariant considering only quadratic order contributions from both position and momentum variables. Next, we obtain two Lewis invariants corresponding to each separable part of the full Hamiltonian of the system. Using both Lewis invariants, one can obtain two Ermakov-Pinney equations, from which we finally obtain the corresponding Lewis phase. Then making an adiabatic approximation enables us to isolate the Berry phase for the full system. After this we obtain some explicit expressions of the Berry phase for a plane polarized gravitational wave with different choices of the harmonic oscillator frequency. Finally, we consider a gravitational wave with cross polarization only interacting with an isotropic two dimensional harmonic oscillator. For this we obtain the Lewis phase and the total Berry phase of the system, which is found to be dependent upon the cross polarization part of the gravitational wave.

Using the Lewis-Riesenfeld′s quantum invariant theory,the Lewis-Riesenfeld phases for a time-dependent frequency harmonic oscillator,which is confined in an infinite well with a moving wall,are calculated.We find that the geometric part of the Lewis-Riesenfeld phases are identical with the “non-adiabatic Berry phases” described by D.Y.Liu in Acta Phys.Sin.It may be important that our results,at least for the system with a sinusoidally oscillating boundary,do not show any non-trivial Berry phase acquired.

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.

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 Lewis' invariant and exact solution for the driven generalized time-dependent harmonic oscillator is found by making use of the Lewis-Riesenfeld quantum theory. Then, the adiabatic asymptotic limit of the exact solution is discussed and the Berry's phase for this system is obtained. We then proceed to use the exact solution to construct the coherent state and calculate the corresponding exact classical phase angle. This phase angle can give the Hannay's angle in the adjabatic limit. The relation between the exact Lewis' phase and the corresponding classical phase angle is discussed.

Abstract The problem of a harmonic oscillator of time dependent frequency confined in a one dimensional infinite square well with a moving wall is studied.It is shown that the exact solution and the Lewis invariant operator of the system can be obtained by performing two consecutive gauge transformations on the time dependent Schrdinger equation.On the basis of the exact solution the Berry phases for the system are calculated by using the geometric concepts such as the geometric distance and geometric length of the curve.